The cost of 12 oranges and 7 apples is $5.36. Eight oranges and 5 apples cost $3.68. Find the
cost of each.

The point estimate for μ is 15.3 years, based on the sample of 60 Grade 9 students.

Based on the statistical techniques given information, the point estimate for the population mean age of all Grade 9 students is 15.3 years. This means that if we assume that the sample is representative of the entire population of Grade 9 students, then we estimate that the average age of all Grade 9 students is 15.3 years.

To estimate the precision of this point estimate, we can calculate confidence intervals. For a 95% confidence interval, we can use the formula:

CI = X ± (Zα/2) * (σ/√n)

where X is the point estimate, Zα/2 is the critical value of the standard normal distribution for a 95% confidence level (1.96), σ is the population standard deviation (which we assume to be known as 4), and n is the sample size (which is 60).

Substituting the values, we get the 95% confidence interval as:

CI = 15.3 ± (1.96) * (4/√60) = (14.33, 16.27)

This means that we can be 95% confident that the true population mean age of Grade 9 students lies between 14.33 and 16.27 years.

For a 99% confidence interval, we can use the same formula with a different value of Zα/2 (2.58 for a 99% confidence level). Substituting the values, we get the 99% confidence interval as:

CI = 15.3 ± (2.58) * (4/√60) = (13.94, 16.66)

This means that we can be 99% confident that the true population mean age of Grade 9 students lies between 13.94 and 16.66 years.

Based on the confidence intervals, we can conclude that the sample provides evidence that the true mean age of all Grade 9 students is likely to be between 14.33 and 16.27 years with a 95% confidence level, and between 13.94 and 16.66 years with a 99% confidence level. However, we cannot be completely certain that the true population mean falls within these intervals as there is always some level of uncertainty associated with sample-based estimates.

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Option C is correct i.e. A+B= (mp + nq)/pq, so the sum of two rational numbers is a rational number.

Given integers are m, n, p and q

And q ≠ 0 and p ≠ 0

A = m / p

B = n/ q

Adding A and B

A + B = m / p + n / q

A + B = (mq + np) / pq

as p ≠ 0 and q ≠ 0 so, pq ≠ 0

So, A + B = (mq + np) / pq is a rational number

Therefore, option C is correct i.e. A+B= (mp + nq)/pq, so the sum of two rational numbers is a rational number.

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Given question is incomplete, the complete question is below:

Keisha's teacher gives her the following information:

• m, n, p, and q are all integers, and p≠ 0 and q≠0

•A= m/q and B= n/p

What conclusion can Keisha make?

A: A +B = (mp + nq)/pq, so the sum of a rational number and an irrational number is an irrational number.

B: A•B= (mp + nq)/pq, so the product of two irrational numbers is an irrational number.

C: A+B= (mp + nq)/pq, so the sum of two rational numbers is a rational number.

D: A•B= (mp + nq)/pq, so the product of two rational numbers is a rational number.​

The ball will rebound to maximum height of 25 centimetres or 0.25 meters after 7 bounces.

Firstly perform the unit conversion. As known, 1 meter is 100 cm. So, 25 centimetres is 0.25 meters.

Now, the formula to be used to find the number of bounces is -

New height × [tex] {2}^{n} [/tex] = old height, where n refers to number of bounces.

Keeping the values in formula

0.25 × [tex] {2}^{n} [/tex] = 32

Rearranging the equation

[tex] {2}^{n} [/tex] = 32/0.25

Divide the values

[tex] {2}^{n} [/tex] = 128

Converting the result into exponent form

[tex] {2}^{n} [/tex] = 2⁷

Thus, n will be 7 bounces.

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Answer:

To find (f + g)(x), we need to add the two functions f(x) and g(x), and then evaluate the sum at x.

So, we have:

(f + g)(x) = f(x) + g(x)

Substituting the given functions, we get:

(f + g)(x) = (2x + 3) + (3x + 2)

Simplifying the expression, we get:

(f + g)(x) = 5x + 5

Therefore, (f + g)(x) is equal to 5x + 5.

E = 0.01(0.0005(30)^2 + 0.38(30) + 35)(2*0.0005(30) + 0.38) ≈ 0.63

Interpretation: The income elasticity of demand is 0.63, which means that for every 1% increase in household income, the fraction of children attending a live musical performance increases by 0.63%.
At a family income level of $30,000, the fraction of children attending a live musical performance is increasing by 0.63% per 1% increase in household income.
To compute the income elasticity of demand E at an income level of $30,000, we need to find the derivative of q with respect to x, and then evaluate it at x=30. The derivative of q with respect to x is:

dq/dx = 0.01(0.001x + 0.38)
Now, let's evaluate this derivative at x=30:
dq/dx = 0.01(0.001(30) + 0.38) = 0.004
To calculate the income elasticity of demand E, we use the formula:
E = (dq/dx)(x/q)
First, let's find q at x=30:
q = 0.01(0.0005(30)^2 + 0.38(30) + 35) = 0.178
Now, we can find E:
E = (0.004)(30/0.178) ≈ 0.67

Interpret the result:
At a family income level of $30,000, the fraction of children attending a live musical performance is increasing by approximately 67% per 1% increase in household income.

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Trapezoids can be split into two triangles and a rectangle.

Trapezoid is also known as a trapezium which is a closed shape having 4  sides with one pair of parallel sides. Trapezium is quadrilateral with 4 sides The parallel sides of a trapezium are known as the bases, and its non-parallel sides are called legs. A trapezium can also have parallel legs. The parallel sides can be horizontal, vertical, or slanting. Few real-life objects example of trapezium is a lamp, popcorn box etc.

A trapezoid consists of two triangles and one rectangle figure shows how can we cut the trapezium to split the trapezium.

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The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.

To find the scale factor of figure KLMN to figure PQRS, given KN = 3 cm, MN = 7 cm, RS = 14 cm, and PS = 6 cm, follow these steps:

1. First, find the length of a side in figure KLMN. We can use KN since it's given: KN = 3 cm.
2. Next, find the corresponding side length in figure PQRS. Since KN corresponds to PS, we have: PS = 6 cm.
3. Now, find the scale factor by dividing the length of the side in figure PQRS by the length of the corresponding side in figure KLMN: scale factor = PS/KN = 6 cm / 3 cm.

The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.

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The total number of students in Mrs. Cunningham's class is 30.

From the question we know that the ratio of boys to girls in Mrs. Cunningham's class is 2 to 3 so we can write

no.of boys: no.of girls = 2:3

The total number of girls in the class is given as 18 so with this we can find out the number of boys in the class that is :

no.of boys= (2/3)*no.of girls in class

now after substituting the values in the equation, we get

no. of boys = (2/3) * 18

no.of boys = 12.

So, now we know the number of boys in the class that is 12 and the number of girls in the class is 18.

We can calculate the total number of students in the class which is equal to

= no.of boys + no.of girls.

= 12+18

=30

Therefore, the total number of students in Mrs. Cunningham's class is 30.

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The area of ΔOPQ is approximately 18.9 square inches, rounding off to nearest 10th.

To find the area of ΔOPQ, we can use the formula:

Area = (1/2) * base * height

We know that p = 9.5 inches, q = 7.6 inches, and ∠O = 31°.

Now, using  trigonometry the height h of the triangle can be found using the sin function.

              sin(θ) = perpendicular/hypotenuse

perpendicular = hypotenuse* sin(θ)

                        = 7.6 * sin(31°)

                        ≈ 3.98 inches

Now, we can use the formula for the area:

Area = (1/2) * base * height

Putting in the values, we get:

Area = (1/2) * 9.5 * 3.98

Area ≈ 18.93 square inches

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A plane rose from take-off and flew at an angle of 11° with the ground, the horizontal distance the plane had flown when it reached an altitude of 500 feet is approximately 2755.3 feet.

To solve this problem, we can use trigonometry. We know that the angle between the ground and the plane's path is 11°, and the altitude of the plane is 500 feet. Let x be the horizontal distance the plane has flown.

We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side of a right triangle, to find x. In this case, the opposite side is the altitude (500 feet) and the adjacent side is x. So we have:

tan(11°) = 500/x

To solve for x, we can multiply both sides by x and then divide by tan(11°):

x = 500 / tan(11°)

Using a calculator, we get:

x ≈ 2755.3 feet

Therefore, the horizontal distance the plane had flown when it reached an altitude of 500 feet is approximately 2755.3 feet.

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Answer:

k=0.3

Step-by-step explanation:

Let's call the length of each of the other two sides x. Since the triangle is isosceles, it has two sides of equal length. Therefore, the perimeter of the triangle can be expressed as 6 + x + x Simplifying this equation, we get  2x + 6 We know that the perimeter is 22 cm so we can set up an equation and solve for x. 22 = 2x + 6 Subtracting 6 from both sides, we get  16 = 2x Dividing both sides by 2, we get x=8

RSQ= 126 degrees

both angles are cooresponding angels therefore

5x+86=10x+46

86-46=10x-5x

40=5x

x=8

substitute

10(x)+46

10(8)+46

80+46

126

If a teacher has an annual salary of 98,500, the teacher makes  $947.12 per biweekly period.

To calculate the teacher's biweekly salary, we need to divide their annual salary by the number of weeks in a year, and then divide that result by 2 (since there are 2 weeks in a biweekly period).

There are a few different ways to approach this calculation, but one common method is to use the following formula:

Biweekly Salary = (Annual Salary / Number of Weeks in a Year) / 2

Using this formula, we can calculate the teacher's biweekly salary as follows:

Biweekly Salary = (98,500 / 52) / 2

Biweekly Salary = (1,894.23) / 2

Biweekly Salary = 947.12

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The correct expressions are:

A) sin θ = y

B) tan θ = y/x

C) cos θ = x

On a unit circle, the terminal side of Angle 0 intersects the circle at points (x,y). We can use trigonometric ratios to relate the coordinates (x,y) to the angle θ.

A) sin θ = the ratio of y to 1, or simply y.

B) tan θ = the ratio of y to x.

C) cos θ = the ratio of x to 1, or simply x.

Therefore, the correct expressions are:

A) sin θ = y

B) tan θ = y/x

C) cos θ = x

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Therefore, the solutions of tan x = -1/2 in the interval (0, 2) are:

x ≈ 2.034, 5.176

We can simplify the given equation as follows:

4 tan 2x - 4 cot x = 0

4(tan 2x - cot x) = 0

4[(2tan x)/(1 - tan^2 x) - (1)/(tan x)] = 0

Multiplying both sides by (1 - tan^2 x) * (tan x), we get:

8tan^3 x - 4tan^2 x - 8tan x + 4 = 0

Dividing both sides by 4 and rearranging, we get:

2tan^3 x - tan^2 x - 2tan x + 1 = 0

Factorizing, we get:

(tan x - 1)(2tan^2 x - tan x - 1) = 0

Using the quadratic formula to solve for the roots of 2tan^2 x - tan x - 1 = 0, we get:

tan x = [1 ± sqrt(1 + 8)] / 4 = [1 ± sqrt(9)] / 4 = 1, -1/2

Therefore, the solutions of the given equation in the interval (0, 2) are the values of x such that tan x = 1 or tan x = -1/2.

We know that tan (π/4) = 1 and tan (-π/4) = -1, so the solutions of tan x = 1 in the interval (0, 2) are:

x = π/4, 5π/4

We can find the solutions of tan x = -1/2 in the interval (0, 2) by finding the reference angle and using the signs of sine and cosine in the corresponding quadrants. We have:

tan x = -1/2

Let θ be the reference angle such that tan θ = 1/2. We know that θ is in the second or fourth quadrant.

In the second quadrant, sine is positive and cosine is negative, so we have:

sin θ = sqrt(1/(1 + tan^2 θ)) = sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = π - θ = π + arctan(1/2) ≈ 2.034

In the fourth quadrant, both sine and cosine are negative, so we have:

sin θ = -sqrt(1/(1 + tan^2 θ)) = -sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = 2π - θ = 2π + arctan(1/2) ≈ 5.176

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The size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.

To determine the function f given that f'(x)=6x² – 5 sin x+eˣ and f(0) = 20, we need to integrate f'(x) with respect to x to obtain f(x), and then use the initial condition f(0) = 20 to find the value of the constant of integration.

Integrating f'(x) with respect to x, we have:

f(x) = 2x³ + 5 cos x + eˣ + C

where C is the constant of integration.

Using the initial condition f(0) = 20, we have:

f(0) = 2(0)³ + 5 cos 0 + e⁰ + C = 6 + C = 20

Therefore, the constant of integration is C = 14, and the function f(x) is:

f(x) = 2x³ + 5 cos x + eˣ + 14

To determine the size of the square that needs to be cut out of each corner of a standard piece of notebook paper to create a box of maximum volume, we can start by drawing a diagram of the box and labeling the sides as follows:

| |

| |

| | h

| |

|__________|

L

Let x be the length of each side of the square that is cut out of each corner. Then, the length and width of the base of the box will be L - 2x and 11 - 2x, respectively, and the height of the box will be x. Therefore, the volume V of the box can be expressed as:

V(x) = x(L - 2x)(11 - 2x)

Expanding and simplifying, we get:

V(x) = -4x³ + 46x² - 110x

To find the size of the square that maximizes the volume of the box, we need to find the value of x that maximizes V(x). This can be done by finding the critical points of V(x) and determining whether they correspond to a maximum or minimum.

Taking the derivative of V(x) with respect to x, we get:

V'(x) = -12x² + 92x - 110

Setting V'(x) = 0 and solving for x, we get:

x = 5/3 or x = 11/6

To determine whether these values correspond to a maximum or minimum, we can use the second derivative test. Taking the second derivative of V(x) with respect to x, we get:

V''(x) = -24x + 92

Evaluating V''(5/3) and V''(11/6), we find that:

V''(5/3) = -4 < 0, so x = 5/3 corresponds to a maximum.

V''(11/6) = 20 > 0, so x = 11/6 corresponds to a minimum.

Therefore, the size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.

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Answer 1: Together, Streetcar and Cable Car are the preferred transportation for 168 residents.

The circle graph shows the percentage of residents who prefer each transportation method, and the total sample size is 400.

For streetcar, (15/100) x 400 = 60 residents prefer it, and for cable car, (27/100) x 400 = 108 residents prefer it.

Together, Streetcar and Cable Car are the preferred transportation for 60 + 108 = 168 residents.

Answer 2: The median is the best measure of center, and it equals 19.

The box plot shows the distribution of the number of tickets sold for a school dance.

The median is the middle value of the data when arranged in order, and it is represented by the line in the box. In this case, the median is 19. The mean, on the other hand, can be influenced by extreme values, and we cannot determine it from the box plot alone.

Answer 3: Median, because Sunny Town is skewed.

When comparing the data, we need to consider the measure of center that is less affected by extreme values, and that is the median.

The median is the middle value of the data when arranged in order. The histogram for Sunny Town is skewed to the right, which means that there are some very high values that are affecting the mean.

Therefore, the median is the better measure of center to determine which location typically has the cooler temperature.

Answer 4: The IQR of 13 is the most accurate to use, since the data is skewed.

The histogram shows the frequency of donations received by a charity, and the data is skewed to the right.

The IQR (Interquartile Range) is the difference between the third quartile (Q3) and the first quartile (Q1), which represents the middle 50% of the data.

The IQR is less sensitive to extreme values and is a better measure of variability for skewed data. In this case, the IQR is 49 - 42 = 7, which is the most accurate measure of variability to use.

Answer 5: There were about 15 principals in attendance.

In the exhibit room, out of 80 people, 15 are principals.

We can assume that the proportion of principals in the exhibit room is the same as the proportion of principals in the conference.

Therefore, the estimated number of principals in the conference is (15/80) x 900 = 168.75, which is approximately 169.

Answer 6: Histogram

The teacher wants to represent the subject preferences of 100 students. A histogram would be the best graphical representation to use because it shows the frequency distribution of a continuous variable, which in this case could be the number of students who prefer each subject.

A stem-and-leaf plot is used for small datasets, and a box plot is used to display the distribution of a continuous variable across categories. A circle graph is more appropriate for displaying categorical data, such as the percentage of students who prefer each subject.

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Answer:

--------------------

Let the original number be n.

We are given that 30% of n is 14.99.

Set up an equation to reflect this relation:

Solve it for n:

The area of the semicircle is approximately 57 square feet.

The arc length of the semicircle can be found using the formula:

arc length = (θ/360) × 2πr

where θ is the angle in degrees, r is the radius, and π is approximately 3.14.

In this case, the diameter of the semicircle is 12 feet, so the radius is half of that, or 6 feet. The angle of the semicircle is 180 degrees, since it is a semicircle. Plugging these values into the formula, we get:

arc length = (180/360) × 2π(6) ≈ 18.85 feet

Therefore, the arc length of the semicircle is approximately 19 feet.

To find the area of the semicircle, we can use the formula:

area = (πr^2)/2

Plugging in the value of the radius from before, we get:

area = (π(6^2))/2 ≈ 56.55 square feet

Therefore, the area of the semicircle is approximately 57 square feet.

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Answer:

C

Step-by-step explanation:

The lowest point on the graph on the y-axis is -4

For approximately 64% of the shows, the theater was half full or less than half full.

Since the seat capacity of the theater is 300, half full would be 150 seats or less. Looking at the histogram, we can see that there are 3 bars representing showings with 150 or less viewers.

The first bar represents showings with 0-50 viewers. From the histogram, it looks like there were about 5 showings with this number of viewers.

The second bar represents showings with 50-100 viewers. From the histogram, it looks like there were about 8 showings with this number of viewers.

The third bar represents showings with 100-150 viewers. From the histogram, it looks like there were about 3 showings with this number of viewers.

So the total number of showings with 150 or less viewers is 5+8+3 = 16.

Since the histogram shows a total of 25 showings, the fraction of shows that was half full or less than half is:

16/25 = 0.64 or 64%

Therefore, for approximately 64% of the shows, the theater was half full or less than half full.

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The volume of the table tent 3 cmH8 cm12 cm is 288 cubic cm.

The volume of the table tent can be found by multiplying the length, width, and height of the tent.

Volume = length x width x height

In this case, the length is 12 cm, the width is 8 cm, and the height is 3 cm.

Volume = 12 cm x 8 cm x 3 cm

Volume = 288 cubic cm

Therefore, the volume of the table tent is 288 cubic cm.

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The circle has horizontal tangent lines at the points (1, 2 + √(14)) and (1, 2 - √(14)) and the circle has vertical tangent lines at the points (1 + √(8), -2) and (1 - √(8), -2).

To find the points where the circle defined by x^2 + y^2 - 2x - 4y = 11 has horizontal and vertical tangent lines, we need to use implicit differentiation to find the derivatives of x and y with respect to each other.

Taking the derivative of both sides of the equation with respect to x, we get:

2x + 2y(dy/dx) - 2 - 4(dy/dx) = 0

Simplifying, we get:

(dy/dx) = (x-1) / (y+2)

To find the points where the circle has horizontal tangent lines, we need to find where the derivative dy/dx is equal to zero. This occurs when x-1 = 0, or x = 1. Substituting this value of x back into the original equation, we get:

1 + y^2 - 4y = 11

Simplifying, we get:

y^2 - 4y - 10 = 0

Using the quadratic formula, we get:

y = 2 ± √(14)

To find the points where the circle has vertical tangent lines, we need to find where the derivative dy/dx is undefined, which occurs when y+2 = 0, or y = -2. Substituting this value of y back into the original equation, we get:

x^2 - 2x + 4 = 11

Simplifying, we get:

x^2 - 2x - 7 = 0

Using the quadratic formula, we get:

x = 1 ± √(8)

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Answer:

Step-by-step explanation

π corresponds to 180 degrees.

so

180 : 8 = 22.5°

The area of the parallelogram is approximately 21.2 square feet, rounded to the nearest tenth of a square foot.

To find the area of a parallelogram, we need to multiply the base by the height. In this case, the two sides given (9.5 feet and 6.1 feet) are the base and height respectively, because they are perpendicular.

To find the area, we first need to find the length of the other two sides. Since a parallelogram has opposite sides that are equal in length, we know that the other two sides are also 9.5 feet and 6.1 feet.

Next, we can use the given angle (20°) to find the height of the parallelogram. We can use trigonometry, specifically the tangent function, to do this:

tan(20°) = height / 6.1 feet

height = 6.1 feet * tan(20°)

height ≈ 2.23 feet

Now we can find the area:

area = base * height

area = 9.5 feet * 2.23 feet

area ≈ 21.185 square feet

Rounding to the nearest tenth of a square foot, the area is approximately 21.2 square feet.

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The volume of the larger box is 200 cubic inches, and the volume of the smaller box is 150 cubic inches.

To find the volume of each box, we use the formula for the volume of a rectangular prism, which is V = lwh, where l is the length, w is the width, and h is the height.

For the larger box, we have l = 8 inches, w = 2.5 inches, and h = 10 inches, so

Volume of large box = = 8 x 2.5 x 10 = 200 cubic inches.

For the smaller box, we have l = 0.75 x 8 = 6 inches, w = 2.5 inches, and h = 10 inches, so

Volume of small box= 6 x 2.5 x 10 = 150 cubic inches.

The difference in the volumes of the two boxes is

Volume of large box - Volume of small box = 200 - 150 = 50 cubic inches.

The units for the volumes are cubic inches, since we are dealing with three-dimensional space.

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Since the loan is at a simple interest rate of 8% for 5 years, the total amount that the mysterious hand will pay is:

Total amount = Principal + Interest

where Principal is the initial amount of the loan, and Interest is the amount of interest charged on the loan.

The interest charged on the loan can be calculated using the simple interest formula:

Interest = Principal x Rate x Time

where Rate is the interest rate per year, and Time is the time period in years.

Plugging in the given values, we get:

Interest = 6600 x 0.08 x 5 = 2640

So the total amount that the mysterious hand will pay is:

Total amount = 6600 + 2640 = 9240

Therefore, the total amount that the mysterious hand will pay for the massage chair is $9,240.

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x² + 10x + 25 is the expression of (x+5)^2 in trinomial in standard form

A trinomial is a polynomial that consists of three terms. It is a type of algebraic expression that contains three algebraic terms separated by either plus or minus signs.

For example, the expression 2x^2 + 5x - 3 is a trinomial because it has three terms: 2x^2, 5x, and -3. Similarly, the expression 4a^3 - 7a^2 + 2a is also a trinomial because it has three terms: 4a^3, -7a^2, and 2a.

We have to expand this

x² + 2(x)(5) + 5²

= x² + 10x + 25

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The runner's average speed is approximately 11.63 mph, which is slightly faster than 11 mph. So, the runner doesn't maintain exactly 11 mph throughout the entire race, but they are close.

If a runner completes a 6.2 mile race in 32 minutes, we can calculate their average speed by dividing the total distance by the total time.

6.2 miles ÷ 32 minutes = 0.194 miles per minute

To convert this to miles per hour, we need to multiply by 60 (the number of minutes in an hour):

0.194 miles per minute x 60 minutes = 11.63 miles per hour

So the runner must be running at an average speed of 11.64 miles per hour throughout the entire race in order to complete it in 32 minutes. This is an impressive pace and shows that the runner is very fit and capable of sustaining a fast speed for a relatively long distance.

A runner completes a 6.2-mile race in 32 minutes, and we are to determine if they maintain an 11 mph pace throughout the race. To do this, we need to convert the race time to hours and then divide the race distance by the time.

First, convert 32 minutes to hours:
32 minutes / 60 minutes/hour = 0.5333 hours

Next, calculate the average speed:
6.2 miles / 0.5333 hours ≈ 11.63 mph

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Answer:

w

Step-by-step explanation:

w