If entire fleet of trucks was normally distributed with a mean of 55 thousand miles and a standard deviation of 11 thousand miles, the z-score for this truck is -1.18. So, correct option is D.
To calculate the z-score for the truck that traveled 42 thousand miles in the last year, we use the formula:
z = (x - μ) / σ
where x is the distance traveled by the truck, μ is the mean distance traveled by all trucks in the fleet, and σ is the standard deviation of the distances traveled by all trucks in the fleet.
Substituting the given values, we get:
z = (42 - 55) / 11 = -1.18
A z-score is a measure of how many standard deviations an observation is away from the mean. In this case, since the z-score is negative, it means that the truck traveled fewer miles than the mean distance traveled by all trucks in the fleet.
The magnitude of the z-score tells us how far away from the mean the observation is in terms of standard deviations. In this case, the truck traveled 1.18 standard deviations below the mean distance traveled by all trucks in the fleet.
So, correct option is D.
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the 95% confidence interval for the weights (in pounds) of a special type of rock ahs found to be 2.62, 6.80. what will be the margin of error
The 95% confidence interval for the weights of the special type of rock is given as (2.62, 6.80). The margin of error can be calculated by finding the difference between the upper and lower bounds of the interval, then dividing by 2.
Margin of error = (6.80 - 2.62) / 2 = 4.18 / 2 = 2.09 pounds
The margin of error for this 95% confidence interval can be calculated as half of the width of the interval.
Width of interval = upper limit - lower limit = 6.80 - 2.62 = 4.18
Therefore, the margin of error will be half of the width of the interval, which is 4.18/2 = 2.09.
So, the margin of error for this 95% confidence interval is 2.09 pounds. This means that if we were to take multiple samples of this special type of rock and construct 95% confidence intervals for each sample, about 95% of the intervals would contain the true population weight of the rock, with a margin of error of 2.09 pounds.
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two linearly independent solutions of the differential equation y'' - 6y' 25y=0
According to differential equation, the two linearly independent solutions are y₁ = e³ᵃcos(4t) and y₂ = e³ᵃsin(4t)
The differential equation y'' - 6y' + 25y = 0 is a second-order linear homogeneous differential equation.
To find the solutions to this differential equation, we can assume a solution in the form of y = e³ᵃ, where r is a constant. Then we take the derivatives of y with respect to t and substitute them into the differential equation. After some algebraic manipulation, we get a quadratic equation in r.
In this specific case, the quadratic equation in r is r² - 6r + 25 = 0. Using the quadratic formula, we get two values of r: r₁ = 3 + 4i and r₂ = 3 - 4i, where i is the imaginary unit. Therefore, the two linearly independent solutions of the differential equation are y₁ = e³ᵃcos(4t) and y₂ = e^(3t)sin(4t), where cos(4t) and sin(4t) are the real and imaginary parts of e^(4it), respectively.
We can verify that these two solutions are indeed linearly independent by showing that no linear combination of them can yield the trivial solution y = 0. This can be done by assuming that y = c₁y₁ + c₂y₂, where c₁ and c₂ are constants, and showing that c₁ = c₂ = 0 is the only solution to this equation.
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Linear programming models have three important properties. They are:
a. optimality, additivity and sensitivity
b. proportionality, additivity, and divisibility
c. optimality, linearity and divisibility
d. divisibility, linearity and nonnegativity
The three important properties of linear programming models are (B) proportionality, additivity, and divisibility, as they allow for the efficient optimization of a linear objective function subject to linear constraints.
Proportionality means that the objective function and constraints are directly proportional to the decision variables, allowing for easy scaling and comparison of solutions.
Additivity means that the objective function and constraints can be expressed as a sum of individual contributions from each decision variable, enabling efficient computation and analysis.
Divisibility means that the decision variables can take on fractional values, allowing for a wide range of feasible solutions and facilitating sensitivity analysis.
Together, these properties make linear programming a powerful tool for solving optimization problems in a variety of fields, from logistics and manufacturing to finance and resource allocation.
Option B holds true.
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Express x = 2t + 6, y = 6t + 3 in the form y = f(x). (Use symbolic notation and fractions where needed.)
The given equations can be expressed in the form y = f(x) as y = 3x - 15.
To express the given equations, x = 2t + 6 and y = 6t + 3, in the form y = f(x), follow these steps:
1. Solve one of the equations for t. Let's choose the equation for x: x = 2t + 6.
2. Subtract 6 from both sides: x - 6 = 2t.
3. Divide by 2 to isolate t: t = (x - 6) / 2.
4. Now, substitute this expression for t into the equation for y: y = 6((x - 6) / 2) + 3.
5. Simplify the equation by multiplying 6 with the term inside the parenthesis: y = 6(x - 6) / 2 + 3.
6. Simplify further by dividing 6 by 2: y = 3(x - 6) + 3.
7. Distribute the 3 to both terms inside the parenthesis: y = 3x - 18 + 3.
8. Combine like terms: y = 3x - 15.
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A parallelogram has sides of length 14.7 cm and 17.3 cm. The longer diagonal has length 25 4 cm Find the angle opposite the longer diagonal. What is the degree measure of the angle opposite the longer diagonal? Round to the nearest tenth as needed)
The measure of the angle that is opposite to the longer diagonal of the parallelogram is approximately 108.1 degrees.
How to determined the angle opposite the longer diagonal of the parallelogram?To find the angle opposite the longer diagonal of the parallelogram, we can use the law of cosines.
Let's label the sides of the parallelogram as follows:
The shorter side is 14.7 cmThe longer side is 17.3 cmThe longer diagonal is 25.4 cmNow, let's use the law of cosines:
c² = a² + b² - 2ab*cos(C)
where c is the length of the longer diagonal, a and b are the lengths of the sides adjacent to the angle C (the angle opposite the longer diagonal), and cos(C) is the cosine of angle C.
Substituting the values we know:
(25.4)² = (14.7)² + (17.3)² - 2(14.7)(17.3)*cos(C)
Simplifying and solving for cos(C):
cos(C) = (14.7²+ 17.3²- 25.4²) / (214.717.3)
cos(C) = -0.3275
Now we can use the inverse cosine (cos⁻¹) function to find the angle C:
C = cos⁻¹(-0.3275)
C ≈ 108.1 degrees
Therefore, the angle opposite the longer diagonal of the parallelogram is approximately 108.1 degrees.
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Andy, a city horticulturalist, is
developing a new park over an old
city lot. (The center of the park
features a koi pond that will cover
1,245 square feet. The remaining
space will need to be covered with
grass seed. If a 50-pound bag of
grass seed covers up to 7,500
square feet, how many bags of
grass seed will Andy need to buy
to seed the rest of the park?
Height: 150 ft
Base: 108 ft
Write the equation in standard form for the circle x2+y2–9=0.
Standard form for this circle is
[tex]\bold{x^2+y^2=9}[/tex]
sqrt(cos(x))*cos(300x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01, sqrt(6-x^2), -sqrt(6-x^2) from -4.5 to 4.5
The graph of the above expression is attached accordingly.
What is the explanation for the above function?
The given function is a combination of three equations defined over the range -4.5 to 4.5.
The first equation is a complex expression involving trigonometric and algebraic functions. The expression is of the form (√(cos(x)) * cos(300x) + √(abs(x)) - 0.7) * (4 - x^2)^0.01.
The second and third equations are simpler, defined as √(6 - x^2) and -√(6 - x^2) respectively. The function produces a 2D plot, where the first equation generates a complex curve with rapid oscillations due to the multiplication of cosines, while the second and third equations generate semi-circles centered at the origin.
The function is interesting due to the complex equation that generates a visually intriguing plot, especially for smaller values of x.
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Full Question:
Although part of your question is missing, you might be referring to this full question:
What is the graph of the expression?
sqrt(cos(x))*cos(300x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01, sqrt(6-x^2), -sqrt(6-x^2) from -4.5 to 4.5
The computer science department is making a new recruitment poster for computer science majors and
would like to include information about the average salary of a software developer on the Treasure Coast.
To estimate the average salary of a software developer, someone from the department collected a random
sample of 10 local software developers. Their salaries are listed below. Assume that the salaries of all
software developers on the Treasure Coast are normally distributed.
Mean, x' = 81,992.2, Sample Standard Deviation is s = 185.44, Margin of Error is E = 223.83, Confidence Interval is (81,768.37, 82,216.03).
Describe Standard Deviation?Standard deviation is a measure of the dispersion or spread of a set of data around its mean. It tells us how much the individual data points deviate from the mean or average value of the data. A low standard deviation indicates that the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.
The standard deviation is calculated by taking the square root of the variance, which is the average of the squared deviations from the mean. It is usually denoted by the symbol σ (sigma) for a population, or s for a sample.
Standard deviation is an important tool in statistics and is widely used in many fields such as finance, economics, and social sciences. It can help us understand the variability of data and make comparisons between different data sets.
To determine the point estimate x' and the sample standard deviation s, we need to find the mean and standard deviation of the given sample.
Mean:
x' = (81,789 + 81,417 + 82,072 + 82,707 + 81,109 + 81,242 + 82,284 + 81,038 + 82,559 + 82,514) / 10
x' = 81,992.2
Sample Standard Deviation:
First, we need to find the sum of the squared differences from the mean.
(81,789 - 81,992.2)² + (81,417 - 81,992.2)²+ (82,072 - 81,992.2)²+ (82,707 - 81,992.2)²+ (81,109 - 81,992.2)²+ (81,242 - 81,992.2)²+ (82,284 - 81,992.2)² + (81,038 - 81,992.2)²+ (82,559 - 81,992.2)²+ (82,514 - 81,992.2)²
= 2,771,040.22
Then, we divide by n-1 and take the square root.
s = √(2,771,040.22 / 9)
s = 185.44
Using a 99% confidence level, the critical value for a two-tailed t-distribution with 9 degrees of freedom is 3.2508.
Margin of Error:
E = t * (s / √(n))
E = 3.2508 * (185.44 / √(10))
E = 223.83
Confidence Interval:
The 99% confidence interval can be calculated as:
x' ± E
81,992.2 ± 223.83
The confidence interval in interval notation is:
(81,768.37, 82,216.03)
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A computer password consists of fourteen characters. Replications are allowed. Part 1 of 5 (a) How many different passwords are possible if each character may be any lowercase letter or digit? Enter your answer in scientific notation with two digit of accuracy after the decimal point. 19 th The possible number of different passwords is 6.14 10 021 Part 2 of 5 (b) How many different passwords are possible if each character may be any lowercase letter? Enter your answer in scientific notation with two digits of accuracy after the decimal point. The possible number of different passwords is 6,45 * 10" Part: 2/5 Part 3 of 5 (c) How many different passwords are possible if each character may be any lowercase letter or digit, and at least one character must be a digit? Enter your answer in scientific notation with two digits of accuracy after the decimal point. The possible number of different passwords is____
Part 1 of 5 (a) The possible number of different passwords consisting of fourteen characters, where each character may be any lowercase letter or digit is 3.61 * 10²⁴.
To find this, there are 26 lowercase letters and 10 digits, totaling 36 possible characters. Since repetitions are allowed, there are 36 options for each of the 14 positions, so the total number of passwords is 36¹⁴, which equals 3.61 * 10²⁴.
Part 2 of 5 (b) The possible number of different passwords consisting of fourteen characters, where each character may be any lowercase letter is 1.11 * 10²⁰.
To find this, there are 26 lowercase letters. Since repetitions are allowed, there are 26 options for each of the 14 positions, so the total number of passwords is 26¹⁴, which equals 1.11 * 10²⁰.
Part 3 of 5 (c) The possible number of different passwords consisting of fourteen characters, where each character may be any lowercase letter or digit, and at least one character must be a digit is 3.25 * 10²⁴.
To find this, subtract the total number of passwords without any digits (1.11 * 10²⁰) from the total number of passwords with lowercase letters and digits (3.61 * 10²⁴). The result is 3.25 * 10²⁴
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Which of the following represents the solutions of the inequality x ≤ −4?
Answer:
The correct graph is the third graph.
Step-by-step explanation:
It is the only graph where the numbers less than or equal to -4 are shaded.
Point A is located at (5,-2) It is translated along (-8,-3) and then rotated reflected across the line y=x. What are the coordinates of its image after the transformations?
The coordinates of the image point C after the given transformations are (-5, -3).
What is the coordinate point called?
The center of the coordinate system (where the lines intersect) is called the origin. The axes intersect when both x and y are zero. The coordinates of the origin are (0, 0).
To apply the given transformations to point A, we need to follow these steps:
Translate the point along vector (-8, -3) to get a new point B.
Reflect the point B across the line y=x to get the final image point C.
Let's first translate point A to get point B. To do this, we add the components of the translation vector to the coordinates of point A:
B = A + (-8, -3)
B = (5, -2) + (-8, -3)
B = (-3, -5)
So, point B is located at (-3, -5).
Next, we reflect point B across the line y=x to get the final image point C. To do this, we switch the x and y coordinates of point B:
C = (y, x)
C = (-5, -3)
Therefore, the coordinates of the image point C after the given transformations are (-5, -3).
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The constant c is unknown in the equation 4x^2 + 12x = -c. Which set of possible values for c gives equations with no real solutions for x?
A. (1.9.18)
B. [-1.0..3)
C. (-9,0.9)
D. (10,20,30)
The set of possible values for c that gives equations with no real solutions for x is (10, 20, 30), which is option D.
We can solve for x in terms of c by first rearranging the equation:
4x² + 12x = -c
4x² + 12x + c = 0
We can then use the quadratic formula to find the solutions for x:
x = (-b ± √(b² - 4ac)) / 2a
Here, a = 4, b = 12, and c = c.
Plugging these values into the formula gives:
x = (-12 ± √(12² - 4(4)(c))) / (2(4))
x = (-3 ± √(9 - c)) / 2
For this equation to have no real solutions for x, the discriminant (the expression inside the square root) must be negative:
9 - c < 0
c > 9
Therefore, the set of possible values for c for x is (10, 20, 30), which is option D.
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Graphs A and B are the result of combining two linear functions, f(x) and g(x). The functions were combined by addition to form h(x), and then by multiplication to form j(x). Graph A Graph B Which statements describe the graphs? Select two options. Graph B represents h(x). Graph B represents j(x). The y-intercepts of both f(x) and g(x) can be 3. The y-intercepts of f(x) and g(x) have opposite signs. The rate of change for both f(x) and g(x) must be negative.
you can change the contents of a stringbuilder object, but you cannot change the contents of a string object. question 2 options: True or False
The required Answer that will help in the selection of the correct option from the given statement is True.
A string builder is considered a class in JAVA API where it provides the mutual sequence of characteristics. It is used for the requirement of dynamic string manipulation. Using it to form new characters in an existing string.
In comparison with string, the following points help in differentiating the ability, function, and purpose of their creation,
The string is immutable that cannot be changed after its creation.Any modification that takes place eventually creates a new set of strings, hence leaving the original or parent string unchanged. Unlike string builder, string can't be modified after it is been created.To learn more about string builder,
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a small bus can hold a maximum of 20 students. let y represent the number of the students
Answer: 40? 20? 60? 80? 100?
Simplify: (4uv^-5)(2u^8v^5)
The expression (4uv⁻⁵) (2u⁸v⁵) can be simplified as 8u⁹.
Given the expression,
(4uv⁻⁵) (2u⁸v⁵)
We have to simplify the given expression.
We have the product rule for exponents that,
xᵃ . xᵇ = xᵃ⁺ᵇ
Using the same rule and the rearranging of the variables,
(4uv⁻⁵) (2u⁸v⁵) = (4 × 2) (u × u⁸) (v⁻⁵ × v⁵)
= 8 (u⁹) (v⁰)
= 8u⁹
Hence the simplified form of the expression is 8u⁹.
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the citizens of a city were asked to choose their favorite type of pet. the circle graph shows how the citizens answered. if 130,000 citizens answered the question, how many chose cat?
Based on the information provided, the circle graph represents the preferences of the citizens for their favorite type of pet. we can calculate the number of citizens who chose cats by multiplying the total number of citizens (130,000) by that percentage. If the graph shows that 40% of citizens chose cats, then the calculation would be 130,000 * 0.40 = 52,000 citizens who chose cats as their favorite pets.
According to the circle graph, we can see that the blue section represents the percentage of citizens who chose cats as their favorite type of pet. To find out the actual number of citizens who chose cats, we need to use the information that 130,000 citizens answered the question.
First, we need to determine what percentage of citizens chose cats. From the graph, it looks like the blue section represents about 35% of the total.
To calculate this percentage as a decimal, we divide 35 by 100:
35/100 = 0.35
Now we can use this decimal to find out how many citizens chose cats:
0.35 x 130,000 = 45,500
Therefore, approximately 45,500 citizens chose cats as their favorite type of pet.
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Write a rule and an equation to fit the pattern in the table. X 1,4,6,7,8, Y 28,31,33,34,35 Describe the relationship in words. The value of y is blank the value of x.
Answer: Y = X + 27
Step-by-step explanation:
1 + 27 = 28
4 + 27 = 31
6 + 27 = 33
7 + 27 = 34
8 + 27 = 35
Convert the following repeating consecutive decimals to fractions. For example, 0.499999... 1/2 (a) 0.7777... (b) -99.9999... (c) 0.09090909... (d) 0.123412341234..
0.123412341234... is equal to the fraction 1234/9999.
(a) To convert 0.7777... to a fraction, let x = 0.7777....
Then, multiplying both sides by 10, we get:
10x = 7.7777...
Subtracting the left-hand sides and the right-hand sides of these two equations, we get:
10x - x = 7.7777... - 0.7777...
Simplifying, we get:
9x = 7
Dividing both sides by 9, we get:
x = 7/9
Therefore, 0.7777... is equal to the fraction 7/9.
(b) To convert -99.9999... to a fraction, let x = -99.9999....
Then, multiplying both sides by -1, we get:
-x = 99.9999...
Dividing both sides by 100, we get:
-x/100 = 0.999999...
Since the decimal 0.999999... is equal to 1, we can rewrite the equation as:
-x/100 = 1
Multiplying both sides by -100, we get:
x = -100
Therefore, -99.9999... is equal to the fraction -100.
(c) To convert 0.09090909... to a fraction, let x = 0.09090909....
Then, multiplying both sides by 100, we get:
100x = 9.090909...
Subtracting the left-hand sides and the right-hand sides of these two equations, we get:
100x - x = 9.090909... - 0.090909...
Simplifying, we get:
99x = 9
Dividing both sides by 99, we get:
x = 1/11
Therefore, 0.09090909... is equal to the fraction 1/11.
(d) To convert 0.123412341234... to a fraction, let x = 0.123412341234....
Then, multiplying both sides by 10000, we get:
10000x = 1234.12341234...
Subtracting the left-hand sides and the right-hand sides of these two equations, we get:
10000x - x = 1234.12341234... - 0.12341234...
Simplifying, we get:
9999x = 1234
Dividing both sides by 9999, we get:
x = 1234/9999
Therefore, 0.123412341234... is equal to the fraction 1234/9999.
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How do the areas of the parallelograms compare? On a coordinate plane, parallelograms A B C D and E F G H are shown. Parallelogram A B C D has points (4, 2), (7, 2), (4, 6), (1, 6). Parallelogram E F G H has points (negative 2, 2), (negative 5, 2), (negative 6, 6), and (negative 3, 6).
The areas of both parallelograms compare in that: They both have the same area
How to find the area of the Parallelogram?The coordinates of Parallelogram ABCD are given as:
A(4, 2), B(7, 2), C(4, 6), D(1, 6).
The coordinates of Parallelogram EFGH are:
E(-2, 2), F(-5, 2), G(-6, 6), and H(-3, 6)
The area of a parallelogram is given by the formula:
Area = base * height
So: To do this, we plot ABCD and EFGH on a grid, then we measure the base and the height of both.
For ABCD:
base = 3
height = 4
Area = 3 * 4 = 12 sq.units
For EFGH, we have same thing:
Area = 3 * 4 = 12 sq.units
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Complete question is:
On a coordinate plane, parallelograms A B C D and E F G H are shown. Parallelogram A B C D has points (4, 2), (7, 2), (4, 6), (1, 6). Parallelogram E F G H has points (negative 2, 2), (negative 5, 2), (negative 6, 6), and (negative 3, 6). How do the areas of the parallelograms compare? The area of parallelogram ABCD is 4 square units greater than the area of parallelogram EFGH. The area of parallelogram ABCD is 2 square units greater than the area of parallelogram EFGH. The area of parallelogram ABCD is equal to the area of parallelogram EFGH. The area of parallelogram ABCD is 2 square units less than the area of parallelogram EFGH.
Use the figure to determine the magnitude A B and c justify your answer
The angles in the lines are as follows:
a = 155 degrees
b = 35 degrees
c = 155 degrees
How to find the angles in a line?When a transversal line is cut by a parallel line, angle relationships are formed such as alternate interior angles, corresponding angles, alternate exterior angles, same side interior angles, vertically opposite angles etc.
Therefore, let's find a, b and c as follows:
b + 155 = 180(sum of angles on a straight line)
b = 180 - 155
b = 35 degrees
Hence,
b + c = 180 (same side interior angles)
c = 180 - 35
c = 155 degrees
a = 155 degrees(vertically opposite angles)
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the mathematical model of a nonlinear dynamic system is given. Follow the procedure outlined in this section to derive the linearized model. ( x1 = x2 – X7 | x2 = 2xz' +1+t *70)=0 x2 (O)=-1
The procedure outlined in this section to derive the linearized model. ( x1 = x2 – X7 | x2 = 2xz' +1+t *70)=0 x2 (O)=-1 is 0.
To derive the linearized model of the given nonlinear dynamic system, we need to follow the following steps:
Step 1: Write the state-space representation of the nonlinear dynamic system.
Given the Equations:
x1 = x2 – X7
x2 = 2xz' +1+t *70
We can write the state-space representation as follows:
x' = f(x,t), where
x = [x1, x2, x3, x4, x5, x6, x7]T
x3 = x4 – x5
x4 = x5x6 + x7
x5 = x2
x6 = 2x1
x7 = 0
f(x,t) = [x2 - x7, 2x1z' + 1 + t * 70, x4 - x5, x5x6 + x7, x2, 2x1, 0]T
Step 2: Compute the Jacobian matrix of the system evaluated at the operating point.
To linearize the system, we need to evaluate the Jacobian matrix of the system at the operating point. The operating point is given as x2(0) = -1. Therefore, we have:
x0 = [-1, -70, 0, 0, -1, -2, 0]T
The Jacobian matrix is given by:
J(x0) = ∂f(x,t)/∂x | x=x0 = [Jij], where
Jij = ∂f(i)/∂x(j)
Using the chain rule, we can compute the elements of the Jacobian matrix as follows:
J11 = ∂f(1)/∂x(1) = 0
J12 = ∂f(1)/∂x(2) = 1
J13 = ∂f(1)/∂x(3) = 0
J14 = ∂f(1)/∂x(4) = 0
J15 = ∂f(1)/∂x(5) = 0
J16 = ∂f(1)/∂x(6) = 0
J17 = ∂f(1)/∂x(7) = -1
J21 = ∂f(2)/∂x(1) = 0
J22 = ∂f(2)/∂x(2) = 0
J23 = ∂f(2)/∂x(3) = 0
J24 = ∂f(2)/∂x(4) = x6
J25 = ∂f(2)/∂x(5) = 1
J26 = ∂f(2)/∂x(6) = 2z'
J27 = ∂f(2)/∂x(7) = 0
J31 = ∂f(3)/∂x(1) = 0
J32 = ∂f(3)/∂x(2) = 0
J33 = ∂f(3)/∂x(3) = 0
J34 = ∂f(3)/∂x(4) = 1
J35 = ∂f(3)/∂x(5) = -1
J36 = ∂f(3)/∂x(6) = 0
J37 = ∂f(3)/∂x(7) = 0
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Find the surface area of the pyramid. Round to the nearest tenth if necessary.
The surface area of the pyramid is 126. 22 in²
How to determine the valueThe formula for calculating the surface area of a triangular pyramid is expressed as;
SA = BA + 1/2(P + s)
Such that the parameters are;
SA is the surface area of the pyramid.BA is the base area of the pyramid.P is the perimeter of the pyramid.s is the slant heightFrom the information
Perimeter = 3(15.75) = 47. 25 in
Base area = 1/2 × 12.25 × 15.75 = 96. 47 in²
Substitute the values
Surface area = 96. 47 + 1/2 (47. 25 + 12.25)
add the values
Surface area = 96. 47 + 1/2 (59.5)
divide the values
Surface area = 126. 22 in²
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If f and g are inverses of each other, what are g(f(x)) and f(g(x)) equal to?
If the functions f and g are inverses of each other, then g(f(x)) = x and f(g(x)) = x
Calculating the values of the composite functionsIf f and g are inverses of each other, then g(f(x)) = x and f(g(x)) = x for all values of x in the domain of the functions.
The reason for this is that the composition of a function with its inverse is equal to the identity function.
To see why, consider g(f(x)). Since f and g are inverses of each other, g(f(x)) is equivalent to g(f(g(y))) for some y in the domain of g.
But since g is the inverse of f, f(g(y)) is equal to y for all y in the domain of g.
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The image of the point ( 3 , − 5 ) (3,−5) under a translation is ( 8 , − 4 ) (8,−4). Find the coordinates of the image of the point ( − 4 , − 6 ) (−4,−6) under the same translation.
Answer:
(1,-5)
Step-by-step explanation:
first find the rule of the first translation
it went from (3,-5) to (8,-4)
meaning 5 was added to the x value to get from 3 to 8 and 1 was added to the y value to get from -5 to -4
so the rule is (x+5, y+1)
apply that to the next point (-4,-6) where -4 is the x value and -6 is the y value
-4+5=1
-6+1=-5
(1,-5) is the coordinate
May you please help me?
Answer:
c
Step-by-step explanation:
A prism has two identical bases where as a pyramid only had one.
suppose p is invertible and a=pbp^-1. solve for b in terms of a
Starting from the given equation, we have:
a = pbp^-1
Multiplying both sides by p on the right:
ap = pb
Multiplying both sides by p^-1 on the left:
p^-1ap = b
So, the solution for b in terms of a is:
b = p^-1ap
Therefore, if p is invertible and a=pbp^-1, we can solve for b by using the above equation, which expresses b in terms of p and a.
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We want to solve for b in terms of a, so we want to isolate b on one side of the equation. This shows that the original equation holds with [tex]b = p^-1ap[/tex], which means that our solution is correct.
To do this, we can use algebraic operations to manipulate the equation.
First, we can multiply both sides of the equation by p on the right:[tex]ap = pb[/tex]
This gives us a new equation that is equivalent to the original equation. Next, we can multiply both sides of the equation by [tex]p^-1[/tex]on the left:[tex]p^-1ap = b[/tex]
This gives us the solution for b in terms of a. We can see that b is equal to [tex]p^-1ap[/tex], which means that b is obtained by multiplying a by p^-1 on the left and p on the right. In other words, b is obtained by transforming a using the inverse of p.
To check that this solution is correct, we can substitute the expression for b back into the original equation and verify that it holds:
[tex]a = pbp^-1[/tex]
[tex]a = p(p^-1ap)p^-1[/tex]
[tex]a = ap^-1p^-1[/tex]
[tex]a = aa^-1[/tex]
[tex]a = a[/tex]
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Use technology or a z-distribution table to find the indicated area.
The scores on a standardized exam are normally distributed with a mean of 450 and a standard deviation of 40.
Approximately 35% of the scores are greater than which score?
Approximately 35% of the scores are greater than 465.4.
How to obtain probabilities using the normal distribution?
The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution of the data-set, depending if the obtained z-score is positive(above the mean) or negative(below the mean).The z-score table is used to obtain the p-value of the z-score, and it represents the percentile of the measure X in the distribution.The mean and the standard deviation for this problem are given as follows:
[tex]\mu = 450, \sigma = 40[/tex]
35% of the scores are greater than the 100 - 35 = 65th percentile, which is X when Z = 0.385, hence:
0.385 = (X - 450)/40
X - 450 = 0.385 x 40
X = 465.4.
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Akeem weighs 140 pounds and hopes to add weight in order to play football in the fall. He hopes to gain between 0.5 and 1.0 pound per week over the next 10 weeks. Akeem will record his progress on the graph below. On the graph, the solid line represents a weigh-gain rate of exactly 1.0 pound per week. The dotted line represents a weight-gain of exactly 0.5 pound per week
The region on the graph that will contain all the points that would represent Akeem's weight-gain progress = region C
The correct answer is an option (H)
Here, on the graph, the solid line represents a weigh-gain rate of exactly 1.0 pound per week.
And the dotted line represents a weight-gain of exactly 0.5 pound per week.
But Akeem's weight gain is more than 1.0 pounds per week. This means that the slope of this progress must be greater than 1
This means that the line representing this progress has greater slope than the solid line on the graph.
This means that the weight- gain progress lies in the region C.
Therefore, the correct answer is an option (H)
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