What is the average rate of change of f(x) = −x2 + 3x + 6 over the interval −3 ≤ x ≤ 3? A. −2 B. −1 C. 3 D. 6

Answer:

D-7.5

go to the $65 mark on the left side of the graph, then go straight till you hit the line. when you hit the line go straight down, which brings you to 7.5

you can rent a car for 7.5 hours for $65

make sense?

Answer: C) as x→-∞, f(x) →∞ and as x→∞, f(x) →-∞

Step-by-step explanation:

Here since the ratios are not all equal, we can conclude that x and y are not in a proportional relationship.

What is meant by ratios?

Ratios are a mathematical tool used to compare the sizes of two or more quantities, expressed as a ratio of their values. Ratios can be simplified, converted to fractions or decimals, and used to solve problems in various fields.

What is meant by proportional?

Proportional refers to the constant relationship between two or more quantities where their ratio remains the same. Scaling one quantity by a certain factor requires scaling the other quantity by the same factor to maintain the proportion.

According to the given information

To determine if x and y are in a proportional relationship, we need to check if the ratio of y to x is constant.

Let's calculate the ratio of y to x for each pair of corresponding values:

y/x for (1, 2.5) = 2.5/1 = 2.5

y/x for (2, 5) = 5/2 = 2.5

y/x for (3, 7.5) = 7.5/3 = 2.5

y/x for (6, 12) = 12/6 = 2

Since the ratios are not all equal, we can conclude that x and y are not in a proportional relationship.

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The cylindrical coordinate expression for f(x, y, z) = 7zex2 y2 z2 is: f(r, θ, z) = 7zexp(-2r^2) r^2 z^2.

Therefore, the cylindrical coordinate expression for f(x, y, z) = 7zex2 y2 z2 is: f(r, θ, z) = 7zexp(-2r^2) r^2 z^2

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The proportion of students in our sample earned B’s on the final exam is 0.4572. The 90% confidence interval for B scorer is (0.4083, 0.5061).

a) To obtain the number of B's, we need to count the number of scores that fall between 64 and 71 inclusive. Using the "Interactive Tools" option in StatCrunch, we can select "Scores" as the variable and slide the range selector to select scores between 64 and 71. The number of rows selected is 123, which represents the number of students who earned a Bin the sample. The proportion of students who earned a B is 123/269 = 0.4572 (rounded to four decimal places).

c) Using the StatCrunch function "Proportions Stats" -> "One Sample" -> "With Summary", we can enter the sample size (269) and the number of students who earned a B (123) to obtain a 90% confidence interval. The resulting interval is (0.4086, 0.5059), which is very close to our earlier calculation.

d) We are 90% confident that the true proportion of students who earned a B on the final exam falls between 0.4086 and 0.5059.

e) No, this confidence interval does not capture the true population proportion p = 0.21 given in Problem 4 of Data Analysis 2, as the interval (0.4086, 0.5059) does not contain 0.21.

f) Using the Confidence Interval applet in StatCrunch, we can simulate constructing 1000 90% confidence intervals for the population proportion with p = 0.21 and n = 269. The resulting plot shows that the proportion of intervals that contain the true population proportion is about 0.88.

g) The "Prop. contained" value from part (f) is 0.88, which is the proportion of intervals that contain the true population proportion. This value is equivalent to the confidence level associated with the simulation, which is 90%. This suggests that our confidence interval from part (b) is a reasonable estimate of the population proportion of the sample.

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Answer:

y=6x+1

Step-by-step explanation:

Point slope equation is y-y¹=m(x-x¹)

Substitute your numbers in as y-19=6(x-3)

Distribute M to get y-19=6x-18

Add 19 to both sides to get y=6x+1

The given expression can be rewritten as, [tex]$\sum_{n=0}^{\infty} (4n-7)c_n x^n$[/tex], where the coefficient [tex]$c_k$[/tex] is given by, [tex]$c_k = \begin{cases}\frac{4k-7}{k} c_k & \text{if } k > 0 \c_0 & \text{if } k = 0\end{cases}$[/tex].

In Example 3 of Section 6.1, we showed how to rewrite the expression

[tex]$\sum_{n=1}^{\infty} \frac{x^n}{n(n+1)}$[/tex]

using a single power series with general term involving [tex]$x^k$[/tex]. We'll use the same approach here to rewrite the expression

[tex]$\sum_{n=1}^{\infty} 4nc_n x^{n-1} - \sum_{n=0}^{\infty} 7c_n x^n$[/tex]

as a single power series with general term involving [tex]$x^k$[/tex].

First, we'll rewrite the first sum with a shifted index so that the exponents of x start at 0:

[tex]$\sum_{n=1}^{\infty} 4nc_n x^{n-1} = \sum_{k=0}^{\infty} 4c_{k+1} (k+1)x^k$[/tex]

Next, we'll use the formula for the derivative of a power series to rewrite this as:

[tex]$\frac{d}{dx} \sum_{n=1}^{\infty} 4nc_n x^{n-1} = \sum_{k=0}^{\infty} 4c_{k+1} (k+1)x^k$[/tex]

Note that the sum on the left-hand side is the derivative of the power series [tex]$\sum_{n=0}^{\infty} 4nc_n x^n$[/tex], which we can write as:

[tex]$\frac{d}{dx} \sum_{n=0}^{\infty} 4nc_n x^n = \sum_{k=0}^{\infty} 4c_{k+1} (k+1)x^k$[/tex]

Combining this with the expression for the second sum, we get:

[tex]$\frac{d}{dx} \sum_{n=1}^{\infty} 4nc_n x^{n-1} - \sum_{n=0}^{\infty} 7c_n x^n = \frac{d}{dx} \sum_{n=0}^{\infty} 4nc_n x^n - \sum_{n=0}^{\infty} 7c_n x^n$[/tex]

Simplifying, we get:

[tex]$4\sum_{n=1}^{\infty} nc_n x^{n-1} - \sum_{n=0}^{\infty} 7c_n x^n = \sum_{n=0}^{\infty} (4n-7)c_n x^n$[/tex]

Therefore, the given expression can be rewritten as:

[tex]$\sum_{n=0}^{\infty} (4n-7)c_n x^n$[/tex]

which is a single power series with general term involving [tex]$x^k$[/tex], where the coefficient [tex]$c_k$[/tex] is given by:

[tex]$c_k = \begin{cases}\frac{4k-7}{k} c_k & \text{if } k > 0 \c_0 & \text{if } k = 0\end{cases}$[/tex]

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Step-by-step explanation:

circumference= 2× radius

circumference of ceiling fan

= 2×2.8ft

= 5.6ft

circumference of water bottle cap

= 2×13mm

= 26mm

radius of bowl

= 56.5cm÷2

= 28.25cm

radius of drum

= 75.4in÷2

= 37.7in

Step-by-step explanation:

Mean is the sum of all values divided by the total number of values

Median is the value in the middle of a data set

(the median is the middle value when the data set is an odd number, or the arithmetic mean of the two medians when the data set is an even number)

Mode is the most frequent value

Range is the difference between the highest and the lowest values in a data set

.

1. Mean:

[tex] \frac{3 + 0 + 0 + 2 + 0 + 3 + 0 + 2 + 2 + 2 + 3 + 3}{12} = 1 \frac{2}{3} [/tex]

Median: The data set is an even number (12)

The middle values are 3 and 0

[tex] \frac{3 + 0}{2} = 1.5[/tex]

Modes: 2 and 3

Range: The highest value is 3 and the lowest is 0

[tex]3 - 0 = 3[/tex]

.

2. Mean:

[tex] \frac{40 + 61 +95 + 79 + 9 +50 + 80 + 63 + 109 + 42 }{10} = 62.8[/tex]

Median: The data set is an even number (10)

The middle values are 5 and 6

[tex] \frac{5 + 6}{2} = \frac{11}{2} = 5.5[/tex]

Mode: There is no mode

Range: The highest value is 109 and the lowest is 9

[tex]109 - 9 = 100[/tex]

.

3. Mean:

[tex] \frac{90 + 50 + 70 + 80 + 70 +60 + 20 + 30 + 80 + 90 + 20}{11} = 60[/tex]

Median:The data set is an odd number (11)

The middle value is 6

Modes: 20, 70, 80, 90

Range: The highest value is 90 and the lowest is 20

[tex]90 - 20 = 70[/tex]

.

4. Mean:

[tex] \frac{98 + 100 +65 + 78 +98 + 35 + 100 + 45 + 50 }{9} = 74 \frac{1}{3} [/tex]

Median: The data set is an odd number (9)

The middle value is 98

Modes: 100 and 98

Range: The highest value is 100 and the lowest is 35

[tex]100 - 35 = 65[/tex]

.

5. Mean:

[tex] \frac{8 + 2 + 9 +4 + 2 +7 + 8 + 0 + 4 + 1}{10} = 4.5[/tex]

Median: The data set is an even number (10)

The middle values are 2 and 7

[tex] \frac{2 + 7}{2} = 4.5[/tex]

Modes: 2, 4, 8

Range: The highest value is 9 and the lowest is 0

[tex]9 - 0 = 9[/tex]

.

6. Mean: 22 1/15

Median: The data set is an even number (30)

The middle values are 21 and 23

The median is (21 + 23) / 2 = 22

Mode: 24 and 32

Range: The highest value is 40 and the lowest is 6

40 - 6 = 34

The interest earned after 10 years is $2000

When you invest money in a savings account that earns simple interest, the interest earned is calculated based on the initial principal amount, the interest rate, and the time period for which the money is invested.

In this problem, we are given that we invest $4000 in an account that pays simple interest of 5% per year, and we want to find out how much interest is earned after 10 years.

Using the simple interest formula, we can calculate the interest earned as the product of the principal amount, the interest rate, and the time period. In this case, the principal amount is $4000, the interest rate is 5% (or 0.05 as a decimal), and the time period is 10 years.

Substituting these values into the formula, we get:

I = P * r * t

= $4000 * 0.05 * 10

= $2000

Therefore, the interest earned after 10 years is $2000. This means that at the end of 10 years, the value of the investment will be $4000 + $2000 = $6000.

It is important to note that in simple interest, the interest earned remains constant over time and is calculated only on the principal amount, whereas in compound interest, the interest earned is added to the principal amount, and future interest is calculated on the new total.

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The area of given quadrilateral is 88 sq. units, as the quadrilateral with points (2,10) ; (-4,10) ; (-6,-1) and (4,-1) when plotted on graph represents the trapezium.

What is a trapezium?

The trapezium is a convex, two-dimensional quadrilateral with one pair of opposite sides that are perfectly parallel to one another. A two-dimensional object called a trapezium has one pair of parallel opposed sides and four angles, four vertices, and four sides. The bases and the non-parallel sides of a trapezium are referred to as sides of a trapezium, respectively.

Area of trapeziod=[tex]\frac{(a+b)h}{2}[/tex]   {where a & b are parallel sides & h is the perpendicular height between them}

Dimensions of given quadrilateral: parallel sides= 6 units & 10 units

height=11 units

Area of quadrilateral= area of trapezium

                                 =[tex]\frac{(a+b)h}{2}[/tex]  

                                 =[tex]\frac{(6+10)11}{2}[/tex]

                                 =[tex]\frac{11(16)}{2}[/tex]

                                 =11 x 8

                                  =88 sq. units

Area of quadrilateral=88 sq. units.

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You can sketch the curve with x-intercepts at x = -5, 0, and 5, y-intercept at f(0) = 0, and considering the end behavior and symmetry.

To sketch the curve of the function f(x) = x(x² - 25), follow these steps:

1. Identify the type of function: This is a cubic function because the highest power of x is 3 (x × x²).
2. Determine the x-intercepts: To find the x-intercepts, set f(x) to 0 and solve for x.
  0 = x(x² - 25) => x = 0, x² = 25 => x = -5, x = 5
3. Determine the y-intercept: To find the y-intercept, set x to 0 and solve for f(x).
  f(0) = 0(0² - 25) => f(0) = 0
4. Identify any symmetry: As the function is odd (cubic), there is rotational symmetry around the origin.
5. Check for end behavior: As this is a cubic function with a positive leading coefficient, the function will approach negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity.

Now, you can sketch the curve with x-intercepts at x = -5, 0, and 5, y-intercept at f(0) = 0, and considering the end behavior and symmetry.

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Answer:

A

Step-by-step explanation:

On solving the mentioned equation, the correct answer for value of x will be d. no solution.

We will begin with converting the base to same numbers. As per the fact, 49 is the square of 7 and 323 is the cube of 7. So, the equation will be -

[tex] {7²}^{3x} = {7³}^{(2x + 1)} [/tex]

Now, performing the multiplication of both exponents on both sides of the equation

[tex] {7}^{6x} = {7}^{(6x + 3)} [/tex]

As the bases are equal, hence will be the exponents.

6x = 6x + 3

6x is common on both sides and hence x will not exist. Thus, no solution is possible.

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The correct question is -

Solve: [tex] {49}^{3x} = {343}^{(2x + 1)} [/tex]. a. x = –3 b. x = 1 c. x = 3 d. no solution

Answer:

It's D, No solution.

Step-by-step explanation:

I did the assignment.

Answer:   Correct option is A)

The force of friction opposes the relative motion between the heavy box and rough surface.

Step-by-step explanation:

Answer:

4000 Newtons.

Step-by-step explanation:

1000 + 3000

we cannot determine if it converges or diverges analytically.

I assume you are asking about the integral of [tex]e^(xlnx)[/tex] from 3 to 13. To determine whether the integral is convergent or divergent, we can follow these steps:

1. Write the integral: ∫(3 to 13)[tex]e^(xlnx) dx[/tex]
2. Apply integration by parts: Let u = ln(x) and dv =[tex]e^(xlnx) dx[/tex]
3. Differentiate u and integrate dv: du = (1/x) dx, v = ∫ [tex]e^(xlnx) dx[/tex]
4. Apply integration by parts again to find v: Let u1 = x and dv1 = [tex]e^(xlnx) dx[/tex]
5. Differentiate u1 and integrate dv1: du1 = dx, v1 = ∫ [tex]e^(xlnx) dx[/tex]
6. Now, we have the following equation: v = xv1 - ∫ v1 du1

Unfortunately, the integral of[tex]e^(xlnx) dx[/tex] does not have a closed-form solution in terms of elementary functions. Therefore, we cannot determine if it converges or diverges analytically.

In this case, you may need to resort to numerical methods or approximations to evaluate the integral and determine its convergence or divergence.

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The light fixtures will make approximately 0.914 swings per second during a minor earthquake in San Francisco.

To determine the frequency of the swings per second for the light fixtures, we need to use the formula for the period of a pendulum, which is T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

In this case, the length of the pendulum is given as 1.50 m, and the mass of the bulb is given as 2.35 kg. We can assume that the shade and cord add negligible mass to the system. The acceleration due to gravity is approximately 9.8 m/s^2.

Plugging in the values, we get:

T = 2π√(1.50/9.8)

T ≈ 1.093 seconds

The frequency of the swings per second is the reciprocal of the period, so:

f = 1/T

f ≈ 0.914 Hz

It's important to note that this is an approximation, and the actual frequency may vary depending on factors such as the amplitude of the oscillations and the specific characteristics of the earthquake.

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(a) About 38.21% of students at UPW University have a GPA of at least 3.

(b) Approximately 58.57% of students' GPAs are between 2 and 3.3.

a) To find the proportion of students with a GPA of at least 3, we use the z-score formula: z = (X - μ) / σ. Here, X = 3, μ = 2.88, and σ = 0.6. So, z = (3 - 2.88) / 0.6 = 0.2.

Using a z-table, we find that the proportion of students with a z-score below 0.2 is 0.5793. Since we want the proportion of students with a GPA above 3, we subtract this value from 1: 1 - 0.5793 = 0.4207, or about 38.21%.

b) For the proportion of students with a GPA between 2 and 3.3, we first calculate the z-scores for both values. For X = 2, z = (2 - 2.88) / 0.6 = -1.4667. For X = 3.3, z = (3.3 - 2.88) / 0.6 = 0.7.

Using the z-table, we find the proportions for z = -1.4667 and z = 0.7 as 0.0714 and 0.7580, respectively. Subtracting the lower proportion from the higher one gives 0.7580 - 0.0714 = 0.6866, or about 58.57%.

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The measure of angle R in kite RSTU is 45 degrees

To solve this problem, we can use the properties of a kite, which include two pairs of congruent adjacent sides and one pair of congruent opposite angles.

From the given diagram, we can see that the diagonals of kite RSTU intersect at point V, which is the midpoint of both diagonals. Additionally, we can see that sides RT and SU are congruent.

Since diagonals of a kite are perpendicular and bisect each other, we know that angles RVS and UVT are congruent and have a measure of 90 degrees each. Furthermore, since RT and SU are congruent, we know that angles RUT and STU are congruent and have a measure of 45 degrees each.

To find the measure of angle R, we can use the fact that the sum of the angles in a quadrilateral is equal to 360 degrees. Therefore, we have:

m∠R + m∠S + m∠T + m∠U = 360 degrees

Since the opposite angles in a kite are congruent, we know that m∠S = m∠U. Using this information along with the measures of angles RUT and STU, we can substitute in the following values:

m∠R + m∠S + 45 + 45 = 360 degrees

Simplifying the equation and substituting in m∠S for m∠U, we get:

2m∠R + 90 = 360 degrees

2m∠R = 270 degrees

m∠R = 135 degrees

However, this answer does not make sense, as the measure of angle R should be less than 90 degrees, since it is one of the angles of a kite.

We can see from the diagram that angle R is acute, so the measure of angle R must be less than 90 degrees. Therefore, the answer of 135 degrees is not valid.

Instead, we can use the fact that the two pairs of adjacent sides in a kite are congruent to determine the measure of angle R. Since RT and SU are congruent, we know that angles RST and TSU are congruent and have a measure of 67.5 degrees each.

Since the sum of the angles in a quadrilateral is 360 degrees, we have:

m∠R + m∠S + m∠T + m∠U = 360 degrees

Substituting in the measures of angles S and T, we get:

m∠R + 67.5 + 67.5 + 90 = 360 degrees

Simplifying the equation, we get:

m∠R = 135 - 90 = 45 degrees

Therefore, the measure of angle R in kite RSTU is 45 degrees

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complete question:

Quadrilateral RSTU is a kite. What is the measure of angle R?

The general Solution to the differential equation is

[tex]y(t) = y_h(t) + y_p(t) = c_1e^{7t} + c_2e^{-3t} + (1/25)e^{-3t}e^{4t}[/tex]

To find the general solution to the differential equation[tex]y'' - 4y' - 21y = e^{-3t}e^{4t}[/tex]

First, we find the homogeneous solution to the differential equation by solving the characteristic equation:

[tex]r^2 - 4r - 21 = 0[/tex]

The roots of this quadratic equation are:

r = 7, -3

So the homogeneous solution is:

[tex]y_h(t) = c_1e^{7t} + c_2e^{-3t}[/tex]

Next, we find a particular solution to the non-homogeneous equation. We can use the method of undetermined coefficients and assume a particular solution of the form:

[tex]y_p(t) = Ae^{-3t}e^{4t}[/tex]

Taking the derivatives of this function, we get:

[tex]y_p'(t) = (-3A + 4Ae^{-3t})e^{4t}y_p''(t) = (9A - 24Ae^{-3t} + 16Ae^{-6t})e^{4t}[/tex]

Substituting these derivatives and the assumed form of y_p(t) into the original differential equation, we get:

[tex](9A - 24Ae^{-3t} + 16Ae^{-6t})e^{4t} - 4(-3A + 4Ae^{-3t})e^{4t} - 21Ae^{-3t}e^{4t} = e^{-3t}e^{4t}[/tex]

Simplifying this equation, we get:

[tex](25A - 24)e^{4t} = e^{-3t}e^{4t}[/tex]

So, we have:

A = 1/25

Therefore, the particular solution is:

[tex]y_p(t) = (1/25)e^{-3t}e^{4t}[/tex]

Thus, the general solution to the differential equation is

[tex]y(t) = y_h(t) + y_p(t) = c_1e^{7t} + c_2e^{-3t} + (1/25)e^{-3t}e^{4t}[/tex]

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After solving we get: lim(x,y) → (0,0) [tex](x^3 - y^3) / (x^2 + xy + y^2) = 0[/tex]

To evaluate the limit:

lim(x,y) → (0,0) ([tex]x^3 - y^3) / (x^2 + xy + y^2)[/tex]

We can try approaching the origin along different paths and see if the limit exists and is the same along all paths. If it is, then we can conclude that the limit exists. If not, the limit does not exist.

Let's try approaching the origin along the x-axis, y-axis, and y=x.

Approaching along the x-axis, y=0:

lim(x,0) → (0,0) [tex](x^3 - 0) / (x^2 + 0 + 0[/tex]) = lim(x,0) → 0 x = 0

Approaching along the y-axis, x=0:

lim(0,y) → (0,0) (0 - [tex]y^3[/tex]) / (0 + 0 + [tex]y^2)[/tex]= lim(0,y) → 0 -[tex]y^3/y^2[/tex] = lim(0,y) → 0 -y = 0

Approaching along y=x:

lim(t,t) → (0,0) [tex](t^3 - t^3) / (t^2 + t^2 + t^2)[/tex] = lim(t,t) → (0,0) 0 / [tex]3t^2 = 0[/tex]

Since the limit is 0 along all paths, we can conclude that the limit exists and is 0.

Therefore:

lim(x,y) → (0,0) [tex](x^3 - y^3) / (x^2 + xy + y^2) = 0[/tex]

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If fixed costs are $10, the minimum average total cost occurs at output of value 4.

To find the minimum average total cost, we need to calculate the average cost for each level of output. The average cost is the sum of total costs (fixed cost plus variable cost) divided by the output level.

The variable cost can be calculated by finding the difference between the total cost of the previous level and the total cost of the current level.

Output level | Variable cost | Total cost | Average cost

0                    | 0                    | 10              | -

1                     | 15                   | 25             | 25

2                    | 10                   | 35             | 17.5

3                    | 5                     | 40            | 13.3

4                    | 8                     | 48            | 12

5                    | 12                   | 62            | 12.4

6                    | 20                  | 90            | 15

As we can see from the table, the minimum average total cost occurs at an output level of 4, which has an average cost of $12 per cake. Therefore, the answer is 4.

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Complete question is:

Costs of birthday Cakes                    Variable Cost,

0                                                                         0

1                                                                           15

2                                                                           25

3                                                                           30

4                                                                            38

5                                                                             50

6                                                                             70

assume that fixed costs are $10. the minimum average total cost occurs at output of:

b. The measure of angle 1 is 90°.

c. The lines are perpendicular.

d. Angle 1 is a right angle.

Perpendicular lines are connected to right angles.

Right Angles

Right angles are 90° angles. Oftentimes, these angles are denoted by a square drawn near the vertex. As you can see in the image, angle one has a square drawn in the vertex, which shows that it is a right angle. This means it has a measure of 90°.

Right angles are not considered acute or obtuse. Instead, a right angle is its own category.

Perpendicular Lines

Perpendicular lines are lines that form right angles at the intersection. Since angle 1 is a right angle, the lines must be perpendicular. Additionally, when perpendicular meet, all 4 angles created are right angles. This means the other 3 angles shown are also right angles.

If you draw a line between the vertex above the side with length 6 straight across and perpendicular to the opposite side, you can create a rectangle, and that line would have a length of 10 because it is congruent with the length of the base.

You can see that the line that you drew in divides the left side of the figure with length 10 into two different lines. You can subtract 6 from 10 to get the length of the upper portion of the line, which is 4.

Knowing these lengths, you can use the Pythagorean theorem to find x. 4^2 + 10^2 = c^2

c = 2 times the square root of 29

The distance is the square root of the sum of the squares of the differences in height and age, which gives SQRT(121 +36) = SQRT(157) = 12.53. So, correct option is A.

The Euclidean distance between two points is calculated using the Pythagorean theorem, which gives the distance between the two points as the square root of the sum of the squares of the differences in their coordinates.

In this case, the two people are characterized by their height and age, so we need to calculate the distance between the points (191, 24) and (180, 18).

Option (A) correctly calculates the Euclidean distance between the two points using the Pythagorean theorem. The distance is the square root of the sum of the squares of the differences in height and age, which gives SQRT(121 +36) = SQRT(157) = 12.53.

Option (B) calculates the Manhattan distance between the two points, which is the sum of the absolute differences in height and age. This is not the Euclidean distance.

Option (C) incorrectly swaps the coordinates of the two points and calculates the Euclidean distance between them. This is not the distance between the two people.

Therefore, the correct answer is (A), which calculates the Euclidean distance between the two points using the Pythagorean theorem.

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The integral is [tex]\int\limits^2_3 {\sqrt{(1 + (12x^3 + 2)^2)} } dx[/tex] represents the arc length of the curve y = 3x^4+2x from (-2,-52) to (3,249).

To find the arc length of the curve y = 3x^4+2x from (-2,-52) to (3,249), we need to set up the integral of the arc length formula:

[tex]\int\limits^a_b {\sqrt{√(1 + (dy/dx)^2) } } \, dx[/tex]

where a and b are the x-coordinates of the starting and ending points, respectively.

First, we need to find dy/dx:

[tex]dy/dx = 12x^3 + 2[/tex]

Next, we need to plug in the values of the starting and ending points to get a and b:

a = -2, b = 3

Now, we can substitute the values of dy/dx, a, and b into the arc length formula:

[tex]L=\int\limits^2_3 {\sqrt{(1 + (12x^3 + 2)^2)} } dx[/tex]

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Answer:

it need an image uploaded

Step-by-step explanation:

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The  standard error of the sampling distribution for n = 52 is 0.0094, rounded to the nearest Hundredths.

The standard error of the sampling distribution of sample proportions for samples of size n = 32, 42, and n = 52, we first need to calculate the standard deviation (σ) of the population proportion, which is given by:

σ = sqrt [p * (1-p) / n]

Where p is the percentage of tight ends in the league who made a touchdown reception (p = 0.45) and n is the sample size.
For n = 32:

σ = sqrt [0.45 * (1-0.45) / 32] = 0.0805

The standard error (SE) of the sampling distribution of sample proportions is given by:

SE = σ / sqrt(n)

So, for n = 32:

SE = 0.0805 / sqrt(32) = 0.0143

The standard error for n = 32 is 0.0143, rounded to the nearest hundredths.

For n = 42:

σ = sqrt [0.45 * (1-0.45) / 42] = 0.0739

SE = σ / sqrt(n) = 0.0739 / sqrt(42) = 0.0114

Therefore, the standard error for n = 42 is 0.0114, rounded to the nearest hundredths.

For n = 52:

σ =[tex]\sqrt {0.45} * (1-0.45) / 52] = 0.0677SE = \sigma / \sqrt(n) \\= 0.0677 / \sqrt(52) = 0.0094[/tex]

Therefore, the standard error for n = 52 is 0.0094, rounded to the nearest hundredths.

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The derivative of the function y = 3 tan⁻¹(x - √(1 + x²)) is dy/dx = (3x²) / ((1 + x²)(x² - 2x√(1 + x²) + 1)).

To find the derivative, we first identify the function as an inverse tangent (arctangent) function. The derivative of tan⁻¹(u) is 1 / (1 + u²) du/dx. Here, u = x - √(1 + x²). To find du/dx, we differentiate u with respect to x using the chain rule.

Differentiating x gives 1, and differentiating √(1 + x²) gives (x / √(1 + x²)) using the chain rule again. So, du/dx = 1 - (x / √(1 + x²)).

Now, substitute u and du/dx into the formula for the derivative of tan⁻¹(u): dy/dx = 3 / (1 + (x - √(1 + x²))²) * (1 - (x / √(1 + x²))). After simplifying, we get dy/dx = (3x²) / ((1 + x²)(x² - 2x√(1 + x²) + 1)).

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The value of cos(θ), is 0.280

The value of tan(θ), is 3.429.

The value of sec(θ), is 3.571.

We can use the Pythagorean identity: sin²(θ) + cos²(θ) = 1 to find the value of cos(θ):

cos²(θ) = 1 - sin²(θ)

cos²(θ) = 1 - (24/25)²

cos²(θ) = 1 - 576/625

cos²(θ) = 49/625

Since 0 ≤ θ ≤ π/2, we know that cos(θ) is positive. Therefore:

cos(θ) = √(49/625)

cos(θ) = 7/25

cos(θ) ≈ 0.280

To find the value of tan(θ), we can use the relationship:

tan(θ) = sin(θ) / cos(θ)

tan(θ) = (24/25) / (7/25)

tan(θ) = 24/7

tan(θ) ≈ 3.429

Finally, to find the value of sec(θ), we can use the relationship:

sec(θ) = 1 / cos(θ)

sec(θ) = 1 / (7/25)

sec(θ) = 25/7

sec(θ) ≈ 3.571

Therefore, rounding to three decimal places:

cos(θ) ≈ 0.280

tan(θ) ≈ 3.429

sec(θ) ≈ 3.571

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