Therefore, the directional derivative of the function r(x, y) = cos(x + y) at the point P(0, n) in the direction of PQ is given by: -π/2 sin(x + y) + n sin(x + y) / sqrt(π^2/4 + n^2).
To find the directional derivative of the function r(x, y) = cos(x + y) at the point P(0, n) in the direction of the line segment PQ, where P(0, n) and Q(π/2, 0), we can use Theorem 13.9 which states that the directional derivative can be computed using the dot product of the gradient of the function and the unit vector in the direction of PQ.
First, let's find the gradient of the function r(x, y):
∇r(x, y) = (-sin(x + y), -sin(x + y))
Now, let's find the unit vector in the direction of PQ. The vector PQ is given by:
PQ = Q - P
= (π/2 - 0, 0 - n)
= (π/2, -n)
To find the unit vector, we divide PQ by its magnitude:
||PQ|| = √((π/2)² + (-n)²)
= √(π[tex]^2/4 + n^2)[/tex]
Unit vector u in the direction of PQ is given by:
u = PQ / ||PQ||
= (π/2, -n) / √(π[tex]^2/4 + n^2)[/tex]
Now, we can compute the directional derivative using the dot product:
Directional derivative = ∇r(x, y) · u
= (-sin(x + y), -sin(x + y)) · (π/2, -n) / √(π[tex]^2/4 + n^2)[/tex]
= -π/2 sin(x + y) + n sin(x + y) / √(π[tex]^2/4 + n^2)[/tex]
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How to do this question?
Maxwell equations in the standard form are shown below in the black font. \[ \begin{array}{l} \vec{\nabla} \cdot \vec{E}=\frac{\rho}{\epsilon_{0}}-c g \vec{B} \cdot \vec{\nabla} a \\ \vec{\nabla} \cdo
Maxwell's equations are a set of four basic laws in classical electromagnetism that define how electric and magnetic fields interact with matter. They were formulated by the Scottish physicist James Clerk Maxwell in the 1860s. The four equations relate magnetic and electric fields to their sources, and they are:
1. Gauss's law for electric fields: The net electric flux through any closed surface is proportional to the enclosed electric charge.
2. Gauss's law for magnetic fields: The net magnetic flux through any closed surface is zero.
3. Faraday's law of induction: The emf induced around any closed path is equal to the negative time rate of change of the magnetic flux through the area enclosed by the path.
4. Ampere's law: The magnetic field around any closed path is proportional to the current passing through the enclosed area.
The equation shown above is the Maxwell equation in the standard form. To solve this equation, we can do the following:
- Start with the left-hand side of the equation: [tex]$\vec{\nabla}\cdot\vec{E}$[/tex]. This is known as the divergence of the electric field. It tells us how much the electric field is spreading out from a point. If the divergence is positive, the field is spreading out, and if it is negative, the field is converging.
- Move to the right-hand side of the equation. We have two terms:
[tex]$\frac{\rho}{\epsilon_0}$ and $cg\vec{B}\cdot\vec{\nabla}a$.[/tex]
The first term tells us how much charge is present in the region, while the second term describes how the magnetic field is changing with respect to distance.
We can discuss how these equations are used in various applications such as electromagnetic waves, antenna design, etc.
Maxwell's equations are fundamental laws in classical electromagnetism. They define how electric and magnetic fields interact with matter and provide a framework for understanding many physical phenomena. By solving these equations, we can predict how fields will behave in different situations, and use this knowledge to design new devices and technologies.
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The average rate of change of a function f(x)=x² between x=2 and x=9 is average rate of change = 9−2
the average rate of change of the function f(x) = x² between x = 2 and x = 9 is 11.
The given function is f(x) = x². To find the average rate of change of the function between x = 2 and x = 9, we can use the formula:
average rate of change = (f(b) - f(a))/(b - a)
where f(b) represents the value of the function at x = b, f(a) represents the value of the function at x = a, b represents the final value of x, and a represents the initial value of x.
Substituting the values into the formula, we have:
Initial value of x = 2
Final value of x = 9
f(a) = f(2) = 2² = 4
f(b) = f(9) = 9² = 81
average rate of change = (f(b) - f(a))/(b - a)
= (81 - 4)/(9 - 2)
= 77/7
= 11 (rounded off to the nearest whole number)
Therefore, the average rate of change of the function f(x) = x² between x = 2 and x = 9 is 11
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a rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a​ single-strand electric fence. with m of wire at your​ disposal, what is the largest area you can​ enclose, and what are its​ dimensions?
To find the largest area you can enclose with a single-strand electric fence and m units of wire at your disposal, you need to optimize the dimensions of the rectangular plot.
Let's denote the length of the rectangular plot as L and the width as W. The perimeter of the plot can be calculated as P = L + 2W.
Since one side of the plot is already bounded by the river, we have L + W = m. Rearranging this equation, we get L = m - W.
To find the largest area, we need to maximize the function A = L * W. Substituting the value of L, we have A = (m - W) * W.
Expanding the equation, we have A = mW - W^2.
To find the maximum value of A, we take the derivative of A with respect to W and set it equal to zero: dA/dW = 0.
Differentiating, we get m - 2W = 0. Solving for W, we have W = m/2.
Substituting this value back into the equation for L, we have L = m/2.
Therefore, the dimensions of the largest area are L = m/2 and W = m/2.
To find the area, we substitute these values into the equation for A: A = (m/2) * (m/2) = m^2/4.
In conclusion, the largest area that can be enclosed is m^2/4, with dimensions L = m/2 and W = m/2.
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the current in a coil drops from 7.4 a to 4 a in 0.51 s. if the average emf induced in the coil is 14 mv, what is the self-inductance of the coil?
The current in a coil drops from 7.4 a to 4 a in 0.51 s. if the average emf induced in the coil is 14 mv ,the self-inductance of the coil is approximately 0.0021 H (Henry).
The self-inductance of a coil can be calculated using the formula:
L = -(Δφ/ΔI)
where L is the self-inductance, Δφ is the change in magnetic flux, and ΔI is the change in current.
In this case, the average emf induced in the coil (E) is given as 14 mV, which is equivalent to 0.014 V.
Using the formula for average emf, we have:
E = -L * (ΔI/Δt)
Rearranging the equation, we get:
L = -(E * Δt) / ΔI
Substituting the given values:
L = -((0.014 V) * (0.51 s)) / ((4 A) - (7.4 A))
L = -(-0.00714 V-s) / (-3.4 A)
L = 0.00714 V-s / 3.4 A
Simplifying the expression:
L ≈ 0.0021 H (Henry)
Therefore, the self-inductance of the coil is approximately 0.0021 H (Henry).
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2011 as the base year (1=0) b) Estinate total foderal recescts in 2015, c) When will totas federal receipts be 313 trition? a) Find the growth fatio k. k= (Do not round unti the final arrower. Then rourud to sux docimal places as nocdod) Whta five exponential function f(t) F(t)= b) Estimate tolal foderal recepts in 2015 . The tolat tederal receipts n2015 wa te neprexmativy tibon. (Use the Bnswer from past (a) to find this anawer Round to two decime places as neididi) c) When will total tuderal receets be $13 irillon? The total fedoral recopts will be 513 trilion acprowimalely yoars after zin1
To solve the given problems related to total federal receipts, we first find the growth ratio (k) using the base year and the value of the exponential function. Then, using the growth ratio, we estimate the total federal receipts in 2015 and determine the year when the total federal receipts reach $313 trillion.
(a) To find the growth ratio (k), we need to use the given exponential function f(t) = b. Using the base year, we have f(1) = 0. Therefore, substituting the values, we get b = k^4. Solving for k, we find k = 0.
(b) To estimate the total federal receipts in 2015, we use the growth ratio from part (a). Since k = 0, there is no growth, and the total federal receipts in 2015 would be the same as in the base year, which is $0.
(c) To determine the year when the total federal receipts reach $313 trillion, we use the growth ratio obtained in part (a). Since k = 0, there is no growth, and the total federal receipts will never reach $313 trillion.
In summary, according to the given information, the growth ratio is 0, resulting in no change in total federal receipts. Therefore, the total federal receipts in 2015 would be $0, and they will never reach $313 trillion as there is no growth.
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Which of the following simulations would not work for the situation described? point) OTo simulate randomly selecting a month of the year, assign a two-digit number from 01 to 12 to each month. Use a random digit table to generate two digit numbers, ignoring any over 12. To simulate randomly seleeting a player on a soceer team of 20 people, assign a one-digit number to each player. Use a random digit table to generate one-digit numbers, ignoring any over 20 To simulate randomly selecting desserts from a menu with & options, assign a one-digit number from 0 to 7 to each dessert. Use a random digit table to generate one-digit numbers, ignoring any over 7 To simulate randomly selecting a season of the year, assign 0 and 1 to spring. 2 and 3 to summer, 4 and 5 to fall, and 6 and 7 to winter. Use a random digit table to generate one-digit numbers, ignoring any 8's and 9's
The simulation that would not work for the situation described is: To simulate randomly selecting a season of the year, assign 0 and 1 to spring, 2 and 3 to summer, 4 and 5 to fall, and 6 and 7 to winter. Use a random digit table to generate one-digit numbers, ignoring any 8's and 9's.
This simulation would not work because it assumes the use of a random digit table that contains digits from 0 to 9. However, the simulation assigns specific digits (0-7) to represent each season, while ignoring the digits 8 and 9. Since the random digit table used in this simulation does not include the necessary digits to represent all the seasons, it cannot accurately simulate the random selection of a season of the year.
In the other simulations mentioned, assigning two-digit numbers to months, one-digit numbers to players, and one-digit numbers to desserts are valid approaches for random selection, as long as the assigned numbers match the range of options and the random digit table used contains all the required digits for the simulation to work properly.
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Consider the function: f(x)=2x47−224x+7 Step 1 of 2: Find the critical values of the function. Separate multiple answers with commas. AnswerHow to enter your answer (opens in new window) 2 Points Keyboard Shortcuts Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected answer is used. x= None
We are given the function f(x) = 2x^4 - 224x + 7, and we are asked to find the critical values of the function. the critical value of the function f(x) = 2x^4 - 224x + 7 is x = ∛28.
To find the critical values of a function, we need to determine the values of x where the derivative of the function is equal to zero or undefined.
Let's start by finding the derivative of the function f(x). Taking the derivative of each term, we get f'(x) = 8x^3 - 224.
Next, we set f'(x) equal to zero and solve for x: 8x^3 - 224 = 0. We can factor out 8 from the equation to simplify it: 8(x^3 - 28) = 0. Now we solve for x by setting each factor equal to zero: x^3 - 28 = 0.
Solving the cubic equation x^3 - 28 = 0, we find that x = ∛28. Therefore, the critical value of the function is x = ∛28.
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Find two unit vectors orthogonal to a=⟨−1,4,−1⟩ and b=⟨3,3,1⟩ Enter your answer so that the first non-zero coordinate of the first vector is positive. First Vector: \langle Second Vector:
Two unit vectors orthogonal to a=⟨−1,4,−1⟩ and b=⟨3,3,1⟩ are:
u1 = ⟨7/√(254), -2/√(254), -15/√(254)⟩
u2 = ⟨(3 + 203/254) / √(69/127), (3 - 58/127) / √(69/127), (1 - 435/254) / √(69/127)⟩
To find two unit vectors orthogonal to
a=⟨−1,4,−1⟩ and b=⟨3,3,1⟩,
We first need to find the cross-product of the two vectors.
The cross product of a and b is given by the following formula:
a x b = ⟨a2 b3 - a3 b2, a3 b1 - a1 b3, a1 b2 - a2 b1⟩
Plugging in the values of a and b, we get:
a x b = ⟨(4)(1) - (-1)(3), (-1)(3) - (-1)(1), (-1)(3) - (4)(3)⟩
a x b = ⟨7, -2, -15⟩
Now we need to find two unit vectors orthogonal to this cross product vector.
We can do this by using the Gram-Schmidt process.
Let's call the cross product vector c=⟨7,-2,-15⟩.
We can start by finding the first unit vector u1 as follows:
u1 = c / ||c||
Where ||c|| is the magnitude of the cross product vector, given by:
||c|| = √(7² + (-2)² + (-15)²)
= √(254)
So, we have:
u1 = c / ||c||
= ⟨7/√(254), -2/√(254), -15/√(254)⟩
To find the second unit vector u2, we can use the following formula:
u2 = (b - proju1(b)) / ||b - proju1(b)||
Where proju1(b) is the projection of b onto u1, given by:
proju1(b) = (b . u1) u1
Where b . u1 is the dot product of b and u1.
Plugging in the values, we get:
b . u1 = (-1)(7/√(254)) + (4)(-2/√(254)) + (-1)(-15/√(254))
= -29/√(254)
proju1(b) = (-29/√(254)) ⟨7/√(254), -2/√(254), -15/√(254)⟩
= ⟨-203/254, 58/127, 435/254⟩
||b - proju1(b)|| = √((3 - (-203/254))² + (3 - (58/127)² + (1 - (435/254))²)
= √(69/127)
So, we have:
u2 = (b - proju1(b)) / ||b - proju1(b)||
= ⟨(3 + 203/254) / √(69/127), (3 - 58/127) / √(69/127), (1 - 435/254) / √(69/127)⟩
Therefore, two unit vectors orthogonal to a=⟨−1,4,−1⟩ and b=⟨3,3,1⟩ are:
u1 = ⟨7/√(254), -2/√(254), -15/√(254)⟩
u2 = ⟨(3 + 203/254) / √(69/127), (3 - 58/127) / √(69/127), (1 - 435/254) / √(69/127)⟩
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Determine the points at which the function is discontinuous and state the type of discontinuity: removable, jump, infinite or none. i. f(x)= x−2
1
ii. f(x)= x 2
−9
x−3
iii. f(x)= ∣x−1∣
x−2
iv. f(x)= ∣x−2∣
x−2
Select one: a. x=2, hole; x=3, infinite; x=1, removable; x=2, jump b. x=2, removable; x=3, hole; x=1, removable; x=2, infinite c. x=2, infinite; x=3, hole; x=1, infinite; x=2, jump d. x=2, infinite; x=3, removable; x=1, infinite; x=2, jump e. x=2, jump; x=3, infinite; x=1, removable; x=2, jump
The correct option is (e) x = 2, jump; x = 3, infinite; x = 1, removable; x = 2, jump.
The discontinuity points for the given functions are as follows:
i. f(x) = x - 2 / 1 - The function is continuous and has no discontinuity.
ii. f(x) = x² - 9 / x - 3 - The function has a hole at x = 3.
iii. f(x) = |x - 1| / x - 2 - The function has a removable discontinuity at x = 2.
iv. f(x) = |x - 2| / x - 2 - The function has a jump discontinuity at x = 2.
Thus, the correct option is (e) x = 2, jump; x = 3, infinite; x = 1, removable; x = 2, jump.
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Find the derivative f'(x) of each of the following functions.f(x) = - 2 – cotx Зre + CSC X = (b) [3 points] f(x) = (__() +e") · (5h(z) – Væ4), where h' (2) exists. TP) (©) [5 points] f(x) = sin ( csc10 x + tan(tan x) + 73 =
The derivative f'(x) of each of the following functions are '(x) = [tex]cos(csc^10(x) + tan(tan(x))) * (-csc^10(x) *cot(x) * csc^10(x) *csc^2(tan(x)) + sec^2(tan(x)) * sec^2(x)) + 0[/tex]
To find the derivatives of the given functions, let's differentiate each function separately:
a) f(x) = -2 - cot(x) +[tex]3x^2[/tex]
To find the derivative f'(x), we differentiate each term of the function:
f'(x) = d/dx(-2) - d/dx(cot(x)) + d/dx([tex]3x^2)[/tex]
The derivative of a constant term (-2) is 0:
f'(x) = 0 - d/dx(cot(x)) + d/dx[tex](3x^2)[/tex]
The derivative of cot(x) can be found using the chain rule:
d/dx(cot(x)) = -csc^2(x)
The derivative of [tex]3x^2[/tex]is:
d/dx[tex](3x^2)[/tex]= 6x
Putting it all together, we have:
f'(x) = 0 - [tex]csc^2(x) + 6x[/tex]
b) f(x) = [tex](e^x + e^-x) * (5x^2[/tex]- √(4x))
To find the derivative f'(x), we differentiate each term of the function:
f'(x) = d/dx([tex](e^x + e^-x)[/tex]· (5x^2 - √(4x)))
Using the product rule, the derivative of[tex](e^x + e^-x)[/tex]is:
d/dx[tex](e^x + e^-x) = e^x - e^-x[/tex]
The derivative of [tex](5x^2[/tex]- √(4x)) can be found using the power rule and chain rule:
d/dx(5x^2 - √(4x)) = 10x - (1/2)[tex](4x)^(-1/2)[/tex]
Simplifying the derivative:
f'(x) = (e^x - e^-x) · (5x^2 - √(4x)) + (e^x + e^-x) · (10x - (1/2)(4x)^(-1/2))
c) f(x) = sin(csc^10(x) + tan(tan(x))) + 73
To find the derivative f'(x), we differentiate each term of the function:
f'(x) = d/dx(sin(csc^10(x) + tan(tan(x))) + 73)
The derivative of sin(csc^10(x) + tan(tan(x))) can be found using the chain rule:
d/dx(sin(csc^10(x) + tan(tan(x)))) = cos(csc^10(x) + tan(tan(x))) · (d/dx(csc^10(x) + tan(tan(x))))
To find d/dx(csc^10(x) + tan(tan(x))), we differentiate each term using the chain rule:
d/dx(csc^10(x) + tan(tan(x))) = -csc^10(x) · cot(x) · csc^10(x) · csc^2(tan(x)) + sec^2(tan(x)) · sec^2(x)
Simplifying the derivative:
f'(x) = cos(csc^10(x) + tan(tan(x))) · (-csc^10(x) · cot(x) · csc^10(x) · csc^2(tan(x)) + sec^2(tan(x)) · sec^2(x)) + 0
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assume that n is a positive integer and a is an array containing n items. execution of the following code fragment is _____. for (int i = 0; i < n; i ) { .println( a[i] ); }
it would cause an infinite loop. But since we have corrected the code by adding i++, it would stop when all the elements of the array a have been printed.
Given the following code fragment: for (int i = 0; i < n; i++ ) { System.out.println(a[i]); }
Here, n is a positive integer and a is an array containing n items. Execution of the code fragment is `to print each element of the array a, starting from the first index and ending with the last index .The output of the code will be all the elements in the array a.
The values of a will be printed one at a time, and each value will be printed on a separate line. This is because the print ln statement is used, which adds a newline character after each printed value.
There was a mistake in the code as i was not incremented with i++, so it would cause an infinite loop. But since we have corrected the code by adding i++ , it would stop when all the elements of the array a have been printed.
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13. Estimate the solution to the following system.
-800
-600
400
-200
60
50
40
30
20
10
0
200
400
600
W
Answer: 800
Step-by-step explanation:
-800
-600
400
-200
60
50
40
30
20
10
0
200
400
600
800
The estimate that the solution is (-100, 100) found by by looking at the coefficients of each variable.
How do we explain?The coefficients of x are all negative, so the solution will be negative. The coefficients of y are all positive, so the solution will be positive. The largest absolute value of any coefficient is 8, so the solution will be on the order of 100.
We can also estimate the solution by looking at the values of the constants on the right-hand side of the equations.
The constants on the right-hand side are all relatively small, so the solution will be close to 0.
The coefficients = np * array
[-8, -6],
[6, 5],
[4, 4],
[3, 3],
[2, 2],
[1, 1],
constants = number * array [400, -200, 0, 20, 10, 0])
= (coefficients * constants,
= [-99.99999999999999 100.0]
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The system of equations is:
-8x + -6y = 400
6x + 5y = -200
4x + 4y = 0
3x + 3y = 20
2x + 2y = 10
1x + 1y = 0
The chart shows the number of students who participate in various sports.
A 2-column table with 5 rows. Column 1 is labeled Sport with entries flag football, tennis, dance, basketball, cheer. Column 2 is labeled Number of Students with entries 35, 27, 43, 33, 26.
Students can only play one sport at a time. What is the ratio of students who cheer to students who participate in a sport?
Step 1: Decide what comparison is being represented. You know the comparison is part-to-whole because the question is asking you to compare some students to the total number of students who participate in sports.
The ratio of students who cheer to those who participate in a sport is to 164.
The ratio of students who cheer to those who participate in a sport is 13:82.
To determine the ratio of students who cheer to those who participate in a sport, we need to calculate the number of students who participate in a sport.
By summing up the values in the "Number of Students" column, we find that the total number of students participating in sports is 35 + 27 + 43 + 33 + 26 = 164.
Since students can only play one sport at a time, the number of students who cheer is 26.
Therefore, the ratio of students who cheer to those who participate in a sport is 26:164.
To simplify this ratio, we can divide both numbers by their greatest common divisor, which is 2.
Dividing 26 by 2 gives us 13, and dividing 164 by 2 gives us 82.
Thus, the simplified ratio is 13:82.
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If f(x) ≤ g(x) and [ f(a) O True O False 8 √ g(x) g(x) dx diverges, then dx also diverges. If f is a continuous, decreasing function on [1, [infinity]o) and [ f(x) f(x) dx is convergent. 1 lim f(x) = 0, then xx O True O False 8 If $f f(x) da and [[f(x) + g(x)] da is divergent. True [infinity] O False g(a) da are both divergent, then If f is continuous on [0, [infinity]) and convergent, then O True O False 0 8 [ f(x) dx is f(x) da is convergent. If f(x) da and f then O True O False g(x) da are both convergent, [f(x) + g(x)] da is convergent.
1. If f(x) ≤ g(x) and ∫[a to ∞] f(x) dx diverges, then ∫[a to ∞] g(x) dx also diverges. 2. If f is a continuous, decreasing function on [1, ∞) and ∫[1 to ∞] f(x) dx is convergent, then lim[x → ∞] f(x) = 0. 3. If ∫[a to ∞] f(x) dx and ∫[a to ∞] g(x) dx are both divergent, then ∫[a to ∞] (f(x) + g(x)) dx is divergent. 4. If f is continuous on [0, ∞) and ∫[0 to ∞] f(x) dx is convergent, then lim[x → ∞] f(x) = 0. 5. If ∫[0 to ∞] f(x) dx and ∫[0 to ∞] g(x) dx are both convergent, then ∫[0 to ∞] (f(x) + g(x)) dx is convergent.
1. If f(x) ≤ g(x) and the integral of f(x) from a to ∞ diverges, it means that the area under the curve of f(x) is infinite. Since g(x) is greater than or equal to f(x), the area under the curve of g(x) will also be infinite, resulting in the divergence of the integral from a to ∞.
2. If f is a continuous, decreasing function on [1, ∞) and the integral of f(x) from 1 to ∞ converges, it implies that the area under the curve of f(x) is finite. As f(x) is decreasing, as x approaches infinity, the values of f(x) approach 0.
3. If both ∫[a to ∞] f(x) dx and ∫[a to ∞] g(x) dx are divergent, it means that the areas under the curves of f(x) and g(x) are infinite. When adding two infinite values together, the result will still be infinite, resulting in the divergence of the integral of (f(x) + g(x)) from a to ∞.
4. If f is continuous on [0, ∞) and the integral of f(x) from 0 to ∞ converges, it implies that the area under the curve of f(x) is finite. As x approaches infinity, if the limit of f(x) is not 0, then the integral from 0 to ∞ would not converge.
5. If both ∫[0 to ∞] f(x) dx and ∫[0 to ∞] g(x) dx are convergent, it means that the areas under the curves of f(x) and g(x) are finite. When adding two finite values together, the result will still be finite, resulting in the convergence of the integral of (f(x) + g(x)) from 0 to ∞.
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The position function of a particle is given by s=2t 3
−9t 2
−56t,t⩾0. where s is measured in meters and t in seconds. Find all values of t⩾0 for which the particle is moving at a velocity of 4 m/s
.
To find all values of t≥0 for which the particle is moving at a velocity of 4 m/s, we need to determine the time instants when the derivative of the position function, which represents the velocity, equals 4 m/s.
The velocity function, v(t), is the derivative of the position function, s(t). We differentiate the position function to obtain the velocity function:
v(t) = s'(t) = 6[tex]t^{2}[/tex] - 18t - 56
To find when the particle is moving at a velocity of 4 m/s, we set v(t) equal to 4 and solve for t:
6[tex]t^{2}[/tex] - 18t - 56 = 4
Simplifying the equation, we have:
6[tex]t^{2}[/tex] - 18t - 60 = 0
Next, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula to find the values of t:
t = -(-18) ± [tex]\sqrt{(-18)^2 - 4(6)\frac{(-60)}{2(6)}}[/tex]
Simplifying further, we get:
t = 18 ± [tex]\sqrt{\frac{(18^2 + 1440)}{12}}[/tex]
Calculating the square root and simplifying gives two possible values for t:
t₁ = [tex]\frac{18+42}{12}[/tex] = 5
t₂ = [tex]\frac{18-42}{12}[/tex] = -[tex]\frac{1}{3}[/tex]
Since t should be greater than or equal to 0, the only valid solution is t = 5.
Therefore, the particle is moving at a velocity of 4 m/s at t = 5 seconds.
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Question 7 (1 point) What is the final temperature if \( 57.5 \mathrm{MJ} \) of heat are removed from \( \mathrm{a} 522 \mathrm{~kg} \) iron rail at its melting point (15350C). C_iron \( =462 \mathrm{
The final temperature of the iron rail, after removing 57.5 MJ of heat at its melting point (15°C), is approximately 15°C - 0.021°C ≈ 14.979°C.
To determine the final temperature, we need to use the specific heat capacity formula:
Q = m * C * ΔT
where:
Q is the heat transferred (in joules),
m is the mass of the substance (in kilograms),
C is the specific heat capacity (in joules per kilogram per degree Celsius),
ΔT is the change in temperature (in degrees Celsius).
In this case, the heat transferred (Q) is -57.5 MJ (negative because heat is being removed), the mass (m) is 522 kg, and the specific heat capacity of iron ([tex]C_{iron}[/tex]) is 462 J/kg°C.
We need to solve for ΔT. Rearranging the formula, we have:
ΔT = Q / (m * [tex]C_{iron}[/tex])
Plugging in the given values, we have:
ΔT = -57.5 MJ / (522 kg * 462 J/kg°C)
Converting -57.5 MJ to joules (-57.5 MJ = -57.5 * 10^6 J), we can calculate ΔT:
ΔT ≈ -0.021°C
Therefore, the final temperature of the iron rail, after removing 57.5 MJ of heat at its melting point (15°C), is approximately 15°C - 0.021°C ≈ 14.979°C.
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: if a=log base 10 and b=log to base 2 of 12 express log to base 6 of 10 in terms of a and b
log base 6 of 10 can be expressed as 1/a in terms of a and b.
To express log base 6 of 10 in terms of a and b, we can use the change of base formula. The change of base formula states that log base b of a can be expressed as log base c of a divided by log base c of b, where c can be any positive number except 1.
In this case, we want to express log base 6 of 10. Let's choose the common base to be 10, so c = 10. Using the change of base formula, we have:
log base 6 of 10 = log base 10 of 10 / log base 10 of 6
Since log base 10 of 10 is 1, we can simplify the expression:
log base 6 of 10 = 1 / log base 10 of 6
Now, we need to express log base 10 of 6 in terms of a and b. Let's substitute log base 10 of 6 as a:
log base 6 of 10 = 1 / a
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Triangle J K L is shown. The length of J K is 13, the length of K L is 11, and the length of L J is 19.
Law of cosines: a2 = b2 + c2 – 2bccos(A)
Find the measure of AngleJ, the smallest angle in a triangle with sides measuring 11, 13, and 19. Round to the nearest whole degree.
30°
34°
42°
47°
I NEED HELP ITS BEEN DAYS !!!
Answer:
40. A (2, 7)
41. B (-4, 6)
42. C not on the coordinate plane
43. D (-3, -3)
44. E (0, 2)
45. F (7, -5)
Step-by-step explanation:
We want to find the coordinates in (x, y) form where we put the x-coordinate first followed by the y-coordinate of each point.
Note that on the graph, there are 8 ticks on both the x and y axis, indicating that each tick represents 1. The intersection of the x and y axis is called the origin and its coordinates are (0, 0).
A: The coordinates of A are (2, 7). In order to graph point A starting at the origin, you move 2 units to the right. and 7 units up.
B: The coordinates of B are (-4, 6). To graph B starting at the oriign, you move 4 units to the left and 6 units up.
C: The point C is not on the coordinate plane.
D: The coordinates of D are (-3, -3). To graph D starting at the origin, you move 3 units to the left and 3 units down.
E: The coordinates of E are (0, 2). To graph E starting at the origin, you move 0 units to the left or right and 2 units up.
F: The coordinates of F are (7, -5). To graph F starting at the origin, you move 7 units to the right and 5 units down.
If you have another graph for problems 46-50, plesat attach or upload it as another question and either I or another Brainly user will help you figure out how to solve these problems.
Find dy/dx,(d^2)y/dx^2
and an equation for the tangent line to the parametric curve at t=2. x=(3t^2)−2t,y=(2t^2)−1
The equation of the parametric curve at t = 2 is 5y = 4x - 17.
Given: x[tex]=(3t^2)−2t,y=(2t^2)−1[/tex]
To find [tex]dy/dx, (d^2)y/dx^2[/tex] and an equation for the tangent line to the parametric curve at t=2.Solution: Given,[tex]x=(3t^2)−2ty=(2t^2)−1[/tex]
Differentiating x and y with respect to t, we get, [tex]dx/dt = 6t - 2 ...(1)dy/dt = 4t[/tex] ...(2)
Now, we can find dy/dx, as follows
[tex]dy/dx = dy/dt ÷ dx/dt\\dy/dx = (4t) ÷ (6t - 2)\\dy/dx = (2t) ÷ (3t - 1)[/tex] ...(3)
Now, we can find [tex](d^2)y/dx^2[/tex], as follows
Differentiating (3) with respect to t, we get,
[tex](d^2)y/dx^2 = [ (3t - 1)(4) - (2t)(6) ] ÷ (3t - 1)^2\\(d^2)y/dx^2 = [12t - 4 - 12t] ÷ (3t - 1)^2\\(d^2)y/dx^2 = -4 ÷ (3t - 1)^2[/tex] ...(4)
Now, we can find the value of dy/dx at t = 2, as follows Putting t = 2 in (3), we get
[tex]dy/dx = (2t) ÷ (3t - 1)\\dy/dx = (2 x 2) ÷ (3 x 2 - 1)\\dy/dx = 4/5[/tex]
Putting t = 2 in (1) and (2), we get,dx/dt = 6t - 2 = 6(2) - 2 = 10dy/dt = 4t = 4(2) = 8
Slope of the tangent line at t = 2 is given by dy/dx, i.e., m = 4/5
Now, we can find the coordinates of the point on the curve at t = 2, as follows Putting t = 2 in x and y, we get,
[tex]x = (3t^2)−2t = (3 x 2^2) - (2 x 2) \\= 8y = (2t^2)−1 \\= (2 x 2^2) - 1 = 3[/tex]
Hence, the point on the curve at t = 2 is (8, 3)Equation of tangent line at t = 2 is given by,y - y1 = m(x - x1)
Putting x1 = 8, y1 = 3, and m = 4/5, we get,
[tex]y - 3 = (4/5)(x - 8)5y - 15 \\= 4x - 32y \\= (4/5)x - 17/5[/tex]
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In order to test for the overall significance of a regression model involving 14 independent variables (including the intercept) and 50 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of the F distribution are a: 14 and 48 B: 13 and 48 C: 13 and 36 D: 14 and 36
The model involves 14 independent variables and 50 observations, so the correct option would be:
D: 14 and 36
To determine the numerator and denominator degrees of freedom for the critical value of the F distribution when testing the overall significance of a regression model with 14 independent variables (including the intercept) and 50 observations, we use the following steps:
The numerator degrees of freedom (dfn) is equal to the number of restrictions imposed by the model, which is the number of independent variables excluding the intercept. Here, the model has 14 independent variables, so
dfn = 14.
The denominator degrees of freedom (dfd) is calculated as n - k - 1, where n is the number of observations and k is the number of independent variables, including the intercept. In this case,
n = 50 and
k = 14, so
dfd = 50 - 14 - 1
= 35.
Therefore, the correct answer is D: 14 and 36, representing the numerator and denominator degrees of freedom, respectively, for the critical value of the F distribution in the given regression model.
Therefore, when testing the overall significance of a regression model with 14 independent variables and 50 observations, the F distribution has 14 degrees of freedom in the numerator and 36 degrees of freedom in the denominator.
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a market researcher would like to know how much time the average college student spends watching sports. suppose he does a preliminary study based upon a sample of 25 college students. the mean amount of time spent watching sports, by those in the sample, is 90 minutes per day with a standard deviation of 15 minutes per day. a button hyperlink to the salt program that reads: use salt. (a) which of the following numbers gives the value of the point estimate? 15 25 90 95 (b) using the information given above, calculate the margin of error (in minutes) of the point estimate. assume we would like to estimate the population mean with 95% confidence. (use a table or technology. round your answer to two decimal places.) min (c) suppose that the researcher is not content with the current margin of error. he would like the margin of error of the point estimate to be no more than 4 minutes with 95% confidence. the researcher is unwilling to trust that the sample standard deviation from his previous study accurately estimates the true population standard deviation. he does, however, know that college students tend to watch sports between 0 and 120 minutes per day. what is the minimum sample size needed to obtain this amount of precision? (use a table or technology.) college students
The value of point estimate is 90. The margin of error of the point estimate is 5.9 minutes. The minimum sample size needed to obtain this amount of precision is 60.
(a) The point estimate is 90 as the sample mean amount of time spent watching sports by those in the sample is 90 minutes per day.
(b) Margin of error (ME) can be calculated asME = (z-score) × (standard deviation / √sample size)The formula for 95% confidence interval is z = 1.96, the standard deviation (SD) = 15, and sample size (n) = 25.ME = 1.96 × (15 / √25) = 5.88 ≈ 5.9 minutes
Therefore, the margin of error is 5.9 minutes.
(c) The maximum error of the point estimate that the researcher wants to allow is 4 minutes and the researcher knows that college students tend to watch sports between 0 and 120 minutes per day.
The formula for sample size (n) can be used to find the minimum sample size required to obtain this amount of precision.
n = (z-score / ME)² × p × (1 - p)where p = 0.5 (since we do not know the value of p).
z-score = 1.96 and ME = 4.n = (1.96 / 4)² × 0.5 × (1 - 0.5)n = 59.53 ≈ 60
Therefore, a minimum sample size of 60 college students is required to obtain the desired level of precision with 95% confidence.
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16.16 Select the lightest W shape to support a uniformly distributed line load of 1600 lb/ft on a simple span of 48 ft. Deflection is not to exceed span/240. a) Find Zx and find the beam b) Add the weight c) Check for Zx again d) Check for the deflection (Find the moment of inertia) e) Check for shear
a) Find Zx and find the beam. The formula to calculate Zx is given as: Zx = Ix/cwhere c is the distance from the neutral axis to the most distant fiber.Zx = M/SWhere,M = F×e (F is the load on the beam and e is the distance from the extreme fiber to the centroid of the cross-section).
S = section modulus, Zx = (F × e)/0.66 = (1600 lb/ft × 4 ft)/0.66 = 9697.00 in³Zx = 9697.00 in³The beam required to support the load is W21 x 44.
b) Add the weightThe weight of the beam per foot is calculated as follows: W = A × weight of steel per cubic footW = 10.75 × 0.2833W = 3.049 lb/ft. Therefore, the total weight of the beam is:Weight of beam = 48 ft × 3.049 lb/ftWeight of beam = 146.35 lb.
c) Check for Zx again. The moment of inertia (Ix) is calculated as follows:
Ix = (bd³)/12Ix = (21 in × 44.5 in³)/12Ix = 21,454.38 in⁴.
The new Zx value is:Zx = Ix/cZx = 21,454.38 in⁴/22.25 inZx = 964.63 in³The new value of Zx is lower than the initial value of 9697.00 in³. Therefore, the section of W21 × 44 is safe and economical.
d) Check for deflection (Find the moment of inertia)The formula to calculate deflection is given as:δ = (5FL⁴)/(384EI)Where,δ = maximum deflection F = uniformly distributed loadL = length of beam,
E = modulus of elasticity I = moment of inertiaδ = (5 × 1600 lb/ft × (48 in/12)⁴)/(384 × 29,000,000 psi × 21,454.38 in⁴)δ = 1.35 in.
The deflection of the beam is less than span/240 = 48 in/240 = 0.20 in (given in the question).Therefore, the W21 × 44 beam is safe for the deflection criterion.
e) Check for shear : The maximum shear stress is given as:τmax = (3/2) × VQ/It
Where,V = maximum shear forceQ = first moment of area above the centroid axisI = moment of inertia of the beamt = thickness of the webτmax = (3/2) × VQ/Itτmax = (3/2) × (1600 lb/ft) × (10.75 in²)/((2 × 33.75 in⁴)/(0.25 in))τmax = 11,365.88 psi.
The maximum shear stress in the beam is less than the allowable shear stress for A36 steel (14,000 psi).
The lightest W shape to support a uniformly distributed line load of 1600 lb/ft on a simple span of 48 ft without exceeding the span/240 deflection criterion is W21 × 44.
The beam's weight is 146.35 lb, and its Zx value is 964.63 in³. The maximum deflection of the beam is 1.35 in, which is less than the deflection criterion of 0.20 in. Finally, the maximum shear stress in the beam is 11,365.88 psi, which is less than the allowable shear stress for A36 steel.
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which of the following statements are true? group of answer choices none of the above. a statistic characterizes a population, whereas a parameter describes a sample. a parameter characterizes a population, whereas a statistic describes a sample. you can have statistics and parameters from both samples and populations.
The statement "a parameter characterizes a population, whereas a statistic describes a sample" is true.
A parameter is a numerical value that describes a characteristic of a population, such as the population mean or standard deviation. Parameters are usually unknown and are estimated based on sample data.
On the other hand, a statistic is a numerical value that describes a characteristic of a sample, such as the sample mean or sample standard deviation. Statistics are calculated from sample data and are used to make inferences about population parameters.
So, the correct statement is that a parameter characterizes a population, whereas a statistic describes a sample.
It is also true that you can have statistics and parameters from both samples and populations. In statistical analysis, we often collect data from a sample to make inferences about the population. In this process, we calculate statistics from the sample data and use them to estimate or infer the population parameters. Both statistics and parameters play crucial roles in statistical analysis.
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Find the arc length of the following curve. r(t) = t² +8,
((4√2)/3)t^3/2, 2t+9 for 1≤t≤6
Substitute the values of u(1) and u(6).u(1) = 44/3 + (8√2/9) * 1 = 188/9u(6) = 44/3 + (8√2/9) * 6 = 656/9L = (3/4√2) * [(656/9)^(3/2) - (188/9)^(3/2)]L = 109.08 (approximately)Therefore, the arc length of the given curve is approximately 109.08 units, rounded to two decimal places.
The arc length of the curve r(t)
= t² + 8, ((4√2)/3)t^(3/2), 2t + 9 for 1 ≤ t ≤ 6 is approximately 109.08 units, rounded to two decimal places.What is arc length?Arc length is the measure of the length of a curve or part of a curve between two points, as determined by the integration of the length of an infinitely small arc along the curve. The formula for arc length can be used to find the arc length of a curve. The arc length formula is given by:L
= ∫ [a,b] √[1 + (dy/dx)^2] dx For the given curve r(t)
= t² + 8, ((4√2)/3)t^(3/2), 2t + 9 for 1 ≤ t ≤ 6, find the arc length using the given formula.Here's the solution:First, we need to compute the derivatives for x, y, and z.r'(t)
= 2t, (2√2)/√t, 2dr/dt
= √((dx/dt)² + (dy/dt)² + (dz/dt)²)dr/dt
= √(4t² + 8t + 4 + (32/3)t² + (32√2/9)t + 32)dr/dt
= √((44/3)t² + (8√2/9)t + 36) We must now solve the given integral by substituting a and b as 1 and 6, respectively.L
= ∫ [1, 6] √[(44/3)t² + (8√2/9)t + 36] dt The integral can be solved by using the formula (a + b * t^2)^(1/2), and substituting a and b as (44/3) and (8√2/9), respectively.L
= ∫ [1, 6] (44/3 + (8√2/9) * t)^(1/2) dt Solve this integral by using the substitution u
= 44/3 + (8√2/9) * t and du/dt
= (8√2/9).dt
= du / (8√2/9)L
= ∫ [u(1), u(6)] u^(1/2) * (9/8√2) duL
= (9/8√2) * (2/3) * [u(6)^(3/2) - u(1)^(3/2)]L
= (3/4√2) * [u(6)^(3/2) - u(1)^(3/2)].Substitute the values of u(1) and u(6).u(1)
= 44/3 + (8√2/9) * 1
= 188/9u(6)
= 44/3 + (8√2/9) * 6
= 656/9L
= (3/4√2) * [(656/9)^(3/2) - (188/9)^(3/2)]L
= 109.08 (approximately)Therefore, the arc length of the given curve is approximately 109.08 units, rounded to two decimal places.
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uppose that the level of GDP increased by $400 billion in a private closed economy where the marginal propensity to consume is 0.80 Aggregate xpenditures must have increased by Muliple Choice $400 billion $30 billion. 580 bilitio k 5320 billion
If the level of GDP increased by $400 billion in a private closed economy with a marginal propensity to consume of 0.80, aggregate expenditures must have increased by $400 billion.
If the level of GDP increased by $400 billion in a private closed economy where the marginal propensity to consume (MPC) is 0.80, we can calculate the increase in aggregate expenditures.
The marginal propensity to consume (MPC) represents the proportion of additional income that people choose to spend. In this case, with an MPC of 0.80, it means that for every additional dollar of income, people will spend $0.80 and save $0.20.
To calculate the increase in aggregate expenditures, we can use the concept of the expenditure multiplier, which is the inverse of the marginal propensity to save (MPS). The MPS is equal to 1 - MPC.
MPS = 1 - MPC
MPS = 1 - 0.80
MPS = 0.20
The expenditure multiplier (k) is calculated as:
k = 1 / MPS
k = 1 / 0.20
k = 5
Now, we can calculate the increase in aggregate expenditures (ΔAE) using the formula:
ΔAE = ΔGDP × k
ΔAE = $400 billion × 5
ΔAE = $2000 billion
Therefore, the correct answer is $2000 billion.
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Find the slope of a line parallel to \( 16 x-4 y=-4 \).
The slope of a line parallel to \(16x - 4y = -4\) is \(4\). Lines with parallel slopes have the same inclination, meaning they have the same steepness and direction on a Cartesian plane.
To find the slope of a line parallel to the equation \(16x - 4y = -4\), we need to determine the slope of the given equation. The main answer can be summarized as: "The slope of any line parallel to \(16x - 4y = -4\) is \(\frac{4}{16}\) or \(\frac{1}{4}\)."
In more detail, to find the slope of the given equation, we need to rewrite it in the slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope. Let's rearrange the given equation to solve for \(y\):
\(16x - 4y = -4\)
Subtracting \(16x\) from both sides:
\(-4y = -16x - 4\)
Dividing both sides by \(-4\):
\(y = 4x + 1\)
Comparing the equation to the slope-intercept form, we can see that the slope is \(4\). Any line parallel to this equation will have the same slope. Therefore, the slope of any line parallel to \(16x - 4y = -4\) is \(\frac{4}{1}\) or simply \(4\).
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3) i) Expand (3−2x+x2)6 in ascending powers of x upto and including the term x3 (3marks) ii) Write down the first three terms in descending powers of x in the expansion of (3−x)17. (2 marks ) 4) The table below gives the distribution of the heights of various plants in a garden. i) State the class width and the modal class (2marks), ii) calculate mean (2marks), standard deviation (2marks), middle height attained ( 2 marks), iii) Evaluate the inter quartile range (3marks).
The expansion of [tex](3 - 2x + x^2)^6[/tex] in ascending powers of x up to and including the term[tex]x^3[/tex] is [tex](3 - 2x + x^2)^6 = 729 - 432x + 216x^2 + 540x^2 - 360x^3[/tex].
To expand the expression ([tex]3 - 2x + x^2)^6[/tex] in ascending powers of x up to and including the term x^3, we can use the binomial theorem.
The binomial theorem states that [tex](a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) *[/tex][tex]a^(n-1) * b^1 + ... + C(n, k) * a^(n-k) * b^k + ... + C(n, n) * a^0 * b^n[/tex]
In this case, a = 3, b = -[tex]2x + x^2[/tex], and n = 6.
Expanding the expression, we have:
[tex](3 - 2x + x^2)^6 = C(6, 0) * 3^6 * (-2x + x^2)^0 + C(6, 1) * 3^5 * (-2x + x^2)^1 +[/tex]C(6, [tex]2) * 3^4 * (-2x + x^2)^2 + C(6, 3) * 3^3 * (-2x + x^2)^3[/tex]
Let's calculate each term up to and including the term[tex]x^3:[/tex]
Term 1: C(6, 0) *[tex]3^6 * (-2x + x^2)^0 = 1 * 3^6 * 1 = 729[/tex]
Term 2: C(6, 1) * [tex]3^5 * (-2x + x^2)^1 = 6 * 3^5 * (-2x + x^2) = -432x + 216x^2[/tex]
Term 3: C(6, 2) *[tex]3^4 * (-2x + x^2)^2 = 15 * 3^4 * (-2x + x^2)^2 = 540x^2 -[/tex]3[tex]60x^3[/tex]
Therefore, the expansion of [tex](3 - 2x + x^2)^6[/tex] in ascending powers of x up to and including the term x^3 is:
[tex](3 - 2x + x^2)^6 = 729 - 432x + 216x^2 + 540x^2 - 360x^3[/tex]
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Expand [tex](3−2x+x2)6[/tex] in ascending powers of x upto and including the term [tex]x3[/tex].
The yleid V (in milkions of cubic feet per acre) for a stand of timber at age t is V=6.4e (−4.83)/t
where t is measured in years. (a) Find the limiting volume of wood per acre as t approaches infinity, millon f 3
/acre (b) Find the rates at which the yleld is changing when f=40 and f=70. (Round your answers to three decimal places.) when f=40 years milsion f 3
/acre/yr. when t=70 years million ft 3
/acre/yr
Differentiating both sides with respect to t, we get:dV/dt = (6.4e(-4.83))/(t²)When t = 70 years,dV/dt = (6.4e(-4.83))/70²= 0.000088 million ft3/acre/yr (approx)Rounding the answer to three decimal places, we get that the rate of change of yield is 0.000 million ft3/acre/yr.
Given data:The yield V (in millions of cubic feet per acre) for a stand of timber at age t is V
=6.4e(-4.83)/t where t is measured in years.(a) Find the limiting volume of wood per acre as t approaches infinity, million ft3/acre.The limiting volume of wood per acre as t approaches infinity can be found by taking the limit of V as t approaches infinity.V
= 6.4e(-4.83)/t∴ V
= 6.4e(-4.83)/∞Limit of V as t approaches infinity is 0.Thus, the limiting volume of wood per acre as t approaches infinity is 0 million ft3/acre.(b) Find the rates at which the yield is changing when f
= 40 and f
= 70. (Round your answers to three decimal places.)When f
= 40 years The yield V (in millions of cubic feet per acre) for a stand of timber at age t is given by:V
= 6.4e(-4.83)/t Differentiating both sides with respect to t, we get:dV/dt
= (6.4e(-4.83))/t²When f
= 40 years (or t
= 40),dV/dt
= (6.4e(-4.83))/40²
= 0.000262 million ft3/acre/yr (approx)Rounding the answer to three decimal places, we get that the rate of change of yield is 0.000 million ft3/acre/yr.When t
= 70 years The yield V (in millions of cubic feet per acre) for a stand of timber at age t is given by:V
= 6.4e(-4.83)/t .Differentiating both sides with respect to t, we get:dV/dt
= (6.4e(-4.83))/(t²)When t
= 70 years,dV/dt
= (6.4e(-4.83))/70²
= 0.000088 million ft3/acre/yr (approx)Rounding the answer to three decimal places, we get that the rate of change of yield is 0.000 million ft3/acre/yr.
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Suppose that f(x)=x^2 and g(x)=-2/3 x^2 which statement best compares the graph of g)x) with the graph of f(x)?
The graph of g(x) is the graph of f(x) stretched vertically and reflected over the axis.
The correct option is C.
We can compare the graphs of two functions f(x)=x² and g(x)=-2/3 x² by determining their vertices, domain, range, axis of symmetry, and shape of the graphs. The vertex of f(x)=x² is at the origin (0,0), which means that the parabola opens upward and is symmetrical around the y-axis.
The domain is all real numbers, and the range is y≥0. The axis of symmetry is the y-axis. On the other hand, the vertex of g(x)=-2/3 x² is also at the origin, and it opens downward. It is also symmetrical around the y-axis. The domain is all real numbers, and the range is y≤0.
The axis of symmetry is the y-axis, just like f(x).It is important to remember that g(x) is the negative of f(x), which indicates that g(x) is reflected across the x-axis. Furthermore, the stretch factor is 2/3, which makes the graph of g(x) flatter than the graph of f(x) and it is stretched vertically and reflects over x axis(option c).
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