13. the area of a rhombus is 484 square millimeters. one diagonal is one-half as long
as the other diagonal. find the length of each diagonal.

By using Mai’s method, the skirt will cost $21.67, By using Elena’s method, the skirt will cost $21 and I think Mai’s method is correct.

(1) We need to find out how much the skirt cost if we use the Mai method. the Mai method is to multiply $30 by 0.80 and then we need to again multiply it with the result which can be given as,

= 30 × 0. 85 × 0. 85

= 21.67

Therefore, By using Mai’s method, the skirt will cost $21.67.

(2) We need to find out how much the skirt cost if we use Elena’s method. Elena’s method is to  multiply $30 by 0. 70 it can be given as,

= 30 × 0. 70

=$21

Therefore, By using Elena’s method, the skirt will cost $21.

(3) I think Mai’s method is correct because she took one 15% discount first and then considered the second discount which is given by the shop. whereas  Elena considered the two discounts at once and calculated it as 30% where the shop offered a 15% discount on the dress and then added a second discount on the purchase cost or bill amount that is why Mai’s method is correct.

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The unit vector in the direction of steepest ascent at P is <-4/sqrt(17), -1/sqrt(17)>,  and the unit vector in the direction of steepest descent at P is <4/sqrt(17), 1/sqrt(17)>.  A vector that points in a direction of no change at P is ⟨-1,1⟩.

To find the direction of steepest ascent/descent at P(-1,1) for f(x,y) = 4x^4 - 4x^2y + y^2 + 9, we need to find the gradient vector evaluated at P and then normalize it to get a unit vector. The gradient vector is given by

grad f(x,y) = <∂f/∂x, ∂f/∂y> = <16x^3 - 8xy, -4x^2 + 2y>

So, at P(-1,1), the gradient vector is

grad f(-1,1) = <16(-1)^3 - 8(-1)(1), -4(-1)^2 + 2(1)> = <-8,-2>

To find the unit vector that gives the direction of steepest ascent, we normalize the gradient vector

||grad f(-1,1)|| = sqrt[(-8)^2 + (-2)^2] = sqrt(68)

So, the unit vector in the direction of steepest ascent at P is

u = (1/sqrt(68))<-8,-2> = <-4/sqrt(17), -1/sqrt(17)>

To find the unit vector that gives the direction of steepest descent, we take the negative of the gradient vector and normalize it

||-grad f(-1,1)|| = ||<8,2>|| = sqrt[8^2 + 2^2] = sqrt(68)

So, the unit vector in the direction of steepest descent at P is

v = (1/sqrt(68))<8,2> = <4/sqrt(17), 1/sqrt(17)>

To find a vector that points in a direction of no change in the function at P, we need to find a vector orthogonal to the gradient vector at P. One such vector is

n = <2,-8>

To see why this works, note that the dot product of the gradient vector and n is

<16x^3 - 8xy, -4x^2 + 2y> . <2,-8> = 32x^3 - 16xy - 4x^2y + 2y^2

Evaluating this at P(-1,1), we get

32(-1)^3 - 16(-1)(1) - 4(-1)^2(1) + 2(1)^2 = 0

So, the vector n is orthogonal to the gradient vector at P and points in a direction of no change in the function.

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These are answers of asked question.

A. To factor out the Greatest Common Factor (GCF) of the expression 3t^4 - 6t^3 - 9t + 12, we need to identify the highest power of t that can be factored out. In this case, the GCF is 3t. So we can rewrite the expression as follows:

3t^4 - 6t^3 - 9t + 12 = 3t(t^3 - 2t^2 - 3) + 3t(4)

The GCF, 3t, is factored out from the first two terms, leaving us with t^3 - 2t^2 - 3. The last term, 12, is divisible by 3t, so it becomes +3t(4). Therefore, the factored form of the expression is:

3t(t^3 - 2t^2 - 3) + 3t(4)

B. To factor the expression g^2 - 5g - 14 using the Distributive Method, we look for two numbers whose product is -14 and whose sum is -5 (the coefficient of the middle term). In this case, -7 and +2 satisfy these conditions. So we can rewrite the expression as follows:

g^2 - 5g - 14 = (g - 7)(g + 2)

Using the Distributive Property, we multiply (g - 7) by (g + 2) to verify the factoring:

(g - 7)(g + 2) = g(g) + g(2) - 7(g) - 7(2) = g^2 + 2g - 7g - 14 = g^2 - 5g - 14

Therefore, the factored form of the expression is:

(g - 7)(g + 2)

C. To factor the expression r^2 - 64, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). In this case, a = r and b = 8, since 8^2 = 64. So we can rewrite the expression as follows:

r^2 - 64 = (r + 8)(r - 8)

Using the difference of squares formula, we can multiply (r + 8) by (r - 8) to verify the factoring:

(r + 8)(r - 8) = r(r) - r(8) + 8(r) - 8(8) = r^2 - 8r + 8r - 64 = r^2 - 64

Therefore, the factored form of the expression is:

(r + 8)(r - 8)

D. To factor the expression 9p^2 - 42p + 49, we look for two numbers whose product is 49 and whose sum is -42 (the coefficient of the middle term). In this case, -7 and -7 satisfy these conditions. So we can rewrite the expression as follows:

9p^2 - 42p + 49 = (3p - 7)(3p - 7)

Using the Distributive Property, we multiply (3p - 7) by (3p - 7) to verify the factoring:

(3p - 7)(3p - 7) = 3p(3p) - 3p(7) - 7(3p) - 7(7) = 9p^2 - 21p - 21p +

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The required equation for the length of the candle is L = 9-0.75t.

Given that a 9-inch candle burns at the rate of 0.75 in per hour we need to determine the equation for the length L of the candle when it burns for t hours,

So, if the candle is burning at the rate of 0.75 in per hour so after burning for t hour the length decreases by 0.75t, from the original length i.e 9 in

So, we can write the equation as =

L = 9-0.75t

Hence the required equation for the length of the candle is L = 9-0.75t.

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(a) With 98% confidence, the true average student loan debt for university students in Canada falls between $25,883.5 and $26,116.5.

(b) No, this does not contradict the reported average of $26,500.

(a) To construct a 98% confidence interval, we can use the formula:

CI = [tex]\bar{x}[/tex] ± z* (σ/√n)

where [tex]\bar{x}[/tex] is the sample mean, σ is the population standard deviation (unknown), n is the sample size, and z* is the z-value from the standard normal distribution that corresponds to the desired level of confidence (98% in this case).

Substituting the given values, we get:

CI = 26,000 ± 2.33 * (500/√64)
CI = 26,000 ± 116.5
CI = (25,883.5, 26,116.5)

Therefore, we can say with 98% confidence that the true average student loan debt for university students in Canada falls between $25,883.5 and $26,116.5.

(b) No, this does not contradict the reported average of $26,500. The reported average is within the confidence interval we calculated, which means that it is a plausible value for the true average. The sample mean of $26,000 is slightly lower than the reported average, but this could be due to random sampling error.

Overall, we cannot make a definitive conclusion about the true average based on this sample alone, but we can say with 98% confidence that it falls within the range of $25,883.5 to $26,116.5.

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The number of people who liked both milk and curd is 500.

Here's how you can calculate it:

- The number of people who liked only milk is 720.

- The number of people who liked only curd is 880.

- The number of people who did not like either milk or curd is 400.

- Let x be the number of people who liked both milk and curd.

- We can use the formula: Total = Group 1 + Group 2 - Both + Neither.

- Substituting the values, we get: 2500 = 720 + 880 - x + 400.

- Solving for x, we get: x = 500.

- Therefore, 500 people liked both milk and curd.

The number of people who liked at least one of the drinks is 2100.

Here's how you can calculate it:

- The number of people who liked only milk is 720.

- The number of people who liked only curd is 880.

- The number of people who liked both milk and curd is 500.

- Therefore, the total number of people who liked at least one of the drink is: 720 + 880 + 500 = 2100.

Answer:

liked both milks: 900

liked at least 1: 2100

Step-by-step explanation:

there are 2500 people

ONLY LIKE 1 type- 1600 people

400 don't like milk

2500-1600=900

2500-400=2100

The value of x is -4.

We have,

(3x + 25) and (6x - 2) makes one angle.

So,

(9x + 23) is the angle.

Now,

It is bisected in two angles.

So,

9x + 23 = 1/2 x (6x - 2)

9x + 23 = 3x - 1

9x - 3x = -1 - 23

6x = -24

x = -4

Thus,

The value of x is -4.

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The volume of the cone is approximately 133.97 cubic inches. So, correct option is A.

The diameter of the base of a cone is 8 inches, which means that the radius is 4 inches (since radius = diameter/2). The height of the cone is twice the radius, which means the height is 2 x 4 = 8 inches.

The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height.

Substituting the values of r and h into the formula, we get:

V = (1/3)π(4²)(8)

V = (1/3)π(16)(8)

V = (1/3)π(128)

V ≈ 133.97 in³

Therefore, correct option is A.

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Considering the following box plot, The interquartile range is a measure of the spread of the middle 50% of the data.

The interquartile range (IQR) is a measure of statistical dispersion that represents the range between the first quartile (Q1) and the third quartile (Q3) in a dataset. It provides a measure of the spread or variability of the middle 50% of the data.

However, explain how to calculate the interquartile range and the percentage of values included in the interquartile range based on a box plot:

(a) To find the interquartile range, you need to calculate the difference between the upper quartile (Q3) and the lower quartile (Q1). In other words, IQR = Q3 - Q1. The interquartile range is a measure of the spread of the middle 50% of the data.

(b) The interquartile range includes 50% of the values in the data set. This means that the other 50% of values lie outside the interquartile range.

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If 75 people chose tulips as their favourite flower then Yelina surveyed 500 people.

What is surveying?

Surveying is a branch of mathematics that deals with the measurement, analysis, and representation of physical features on land or in space. It uses geometry, trigonometry, and infinitesimal calculus to accurately measure distances, angles, elevations, and positions. Surveying is used in various fields such as civil engineering, construction, mapping, and geology to obtain valuable information about the earth's surface and to create maps and plans for various purposes.

According to the given information

Since the central angle for tulips is 54°, we can set up a proportion:

54/360 = 75/x

Simplifying this proportion, we get:

0.15 = 75/x

Multiplying both sides by x, we get:

x * 0.15 = 75

Dividing both sides by 0.15, we get:

x = 500

Therefore, Yelina surveyed 500 people.

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Answer:

a) £54

b) £1908

Step-by-step explanation:

a) use £1800 × 3% = £54

b) use £1800 + ( 54 × 2) = £1908

To find the volume (cubic inches) of the ball, we first need to find its radius, which is half the diameter.

Radius = 12 in / 2 = 6 in

Now we can use the formula for the volume of a sphere:

Volume = (4/3) x π x radius^3 cubic inches

Volume = (4/3) x π x 6^3

Volume ≈ 904.8 cubic inches

So the ball holds approximately 904.8 cubic inches of air. Rounded to the nearest tenth, the answer is 904.8.

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the roots of the function are[tex]x = 1[/tex] and  [tex]x = 3.[/tex] These are also the x-intercepts of the parabola.

This polynomial is a quadratic function and can be graphed as a parabola. a polynomial of degree two.

[tex]f(x) = x^2 - 4x + 3[/tex]

The coefficient of the x^2 term is positive, which means that the parabola opens upwards. The vertex of the parabola is at (2, -1) and the y-intercept is at (0, 3).

The roots of the function can be found by setting f(x) = 0 and solving for x:

[tex]x^2 - 4x + 3 = 0[/tex]

[tex](x - 3)(x - 1) = 0[/tex]

[tex]x = 1, 3[/tex]

Therefore, the roots of the function are[tex]x = 1[/tex] and  [tex]x = 3.[/tex] These are also the x-intercepts of the parabola.

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Since we have shown that all four angles formed by the lateral sides are right angles, and the opposite sides are parallel and congruent, we can conclude that the lateral sides are rectangles.

Yes, we can prove that the lateral sides are rectangles based on the given information.

Firstly, we can see that the two diagonals of the lateral sides are congruent (both measure 13 cm), which means that the opposite sides of the figure are parallel. This is because, in a rectangle, opposite sides are parallel and congruent.

Next, we can use the converse of the Pythagorean theorem to determine if the angles are right angles. The converse of the Pythagorean theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

For each of the lateral sides of the figure, we can consider the two triangles formed by one of the diagonals and the adjacent sides. Applying the Pythagorean theorem, we can see that:

For the first lateral side, we have:

(5 cm)^2 + (12 cm)^2 = (13 cm)^2

Therefore, the angles between the 5 cm and 12 cm sides are right angles.

For the second lateral side, we have:

(5 cm)^2 + (12 cm)^2 = (13 cm)^2

Therefore, the angles between the 5 cm and 12 cm sides are also right angles.

Since we have shown that all four angles formed by the lateral sides are right angles, and the opposite sides are parallel and congruent, we can conclude that the lateral sides are rectangles.

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Mario can afford to buy a maximum of 12 hamburgers from the local store.

Let's assume Mario can buy "h" hamburgers.

The cost of each hamburger is $2.90, and Mario wants to buy "h" hamburgers, so the total cost of hamburgers would be 2.90h.

He is also buying one packet of hamburger rolls, which costs $4.75.

Therefore, the total amount he can spend at the store must be less than or equal to his budget of $39.55.

Putting it all together, the inequality representing this situation is:

2.90h + 4.75 ≤ 39.55

To find out how many hamburgers Mario can afford to buy, we need to solve the inequality:

2.90h + 4.75 ≤ 39.55

Subtracting 4.75 from both sides of the inequality:

2.90h ≤ 34.80

Next, divide both sides of the inequality by 2.90:

h ≤ [tex]\frac{34.80 }{ 2.90}[/tex]

Simplifying the right side:

h ≤ 12

Therefore, Mario can afford to buy a maximum of 12 hamburgers from the local store.

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The radius of the circle is 10 inches.

A sector is a part of a given circle which is made from two radii and an arc. It's area can be determined by;

area of a sector = θ/360*πr^2

where θ is the measure of its central angle, and r is its radius.

Then from the given question, we have;

area of a sector = θ/360*πr^2

162 = 180/360 *3.14*r^2

      = 1.57r^2

r^2 = 162/1.57

     = 103.1847

So that;

r = (103.1847)^1/2

 = 10.16

The radius of the circle is approximately 10 inches.

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Let's assume that the hourly rate for loading and unloading is $x and the hourly rate for packing and unpacking is $y.

From the given information, we can form the following two equations:

[tex]8x + 6y = 890[/tex] ...(1) (for 8 hours of loading and unloading and 6 hours of packing and unpacking)

[tex]5x + 3y = 515[/tex] ...(2) (for 5 hours of loading and unloading and 3 hours of packing and unpacking)

To solve for x and y, we can use the method of elimination.

Multiplying equation (2) by 2, we get:

[tex]10x + 6y = 1030[/tex] ...(3)

Now, subtracting equation (1) from equation (3), we get:

2x = 140

Therefore, x = $70 per hour.

Substituting the value of x in equation (2), we get:

5(70) + 3y = 515

Simplifying, we get:

3y = 165

Therefore, y = $55 per hour.

Hence, the company's hourly rates are $70 per hour for loading and unloading and $55 per hour for packing and unpacking.

In summary, we can set up a system of equations to solve for the hourly rates of a moving team.

From there, using the method of elimination, we can solve for the hourly rates for both loading and unloading as well as packing and unpacking.

In this case, the hourly rate for loading and unloading is $70 per hour, and the hourly rate for packing and unpacking is $55 per hour.

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The distance between the points P(-4, 2, 1) and Q(-1, 2, 0) is [tex]\sqrt{(10)}[/tex] units.

The distance formula is derived from the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the legs (the sides that form the right angle) is equal to the square of the length of the hypotenuse (the side opposite the right angle).

In three-dimensional space, we have to use a variation of the Pythagorean theorem that involves finding the distance between the two points in each of the three dimensions (x, y, and z) and then adding up the squares of those distances, before taking the square root of the sum.

To find the distance between two points P(x1, y1, z1) and Q(x2, y2, z2) in three-dimensional space, we use the distance formula:

d = [tex]\sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2)}[/tex]

Using the given points P(-4, 2, 1) and Q(-1, 2, 0), we have:

d = [tex]\sqrt{((-1 - (-4))^2 + (2 - 2)^2 + (0 - 1)^2)}[/tex]

= [tex]\sqrt{(3^2 + 0^2 + (-1)^2)}[/tex]

= [tex]\sqrt{(10)}[/tex]

Therefore, the distance between the points P(-4, 2, 1) and Q(-1, 2, 0) is sqrt(10) units.

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The mean winnings for all of the show's contestants can be estimated with 92% certainty. 35014.48385 is the lower bound, while 40669.38281 is the upper bound.            

Lower Bound = [tex]X - t(\alpha/2) * s / \sqrt{(n)[/tex]            

Upper Bound = [tex]X + t(\alpha/2) * s / \sqrt{(n)[/tex]

where                

[tex]\alpha/2 = (1 - confidence\: level)/2 = 0.04 \\ X = sample\: mean = 37841.93333 \\ t(\alpha/2) = critical\: t \:for \:the\: confidence\: interval = 1.887496145 \\ s = sample\: standard\: deviation = 5801.688541 n = sample\: size = 15 \\ df = n - 1 = 14[/tex]          

Thus,          

Lower bound =     35014.48385            

Upper bound =     40669.38281            

A lower bound refers to the smallest possible value or limit that a given quantity or parameter can take. In various fields of mathematics and computer science, lower bounds are used to establish limits on the performance of algorithms, the complexity of computational problems, and the amount of resources required to solve a problem. This information can be useful in developing more efficient algorithms or determining the practicality of a given approach.

Lower bounds are useful for understanding the fundamental limits of a system or process. By establishing a lower bound, researchers and practitioners can better understand the potential of a given technology or approach, and can work to optimize it within the constraints imposed by the lower bound.

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Complete Question:-

How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and listed below. Estimate with 92% confidence the mean winning's for all the show's players.

30692 43231 48269 28592 28453

36309 45318 36362 42871  39592

35456 40775 36466 36287 38956

Lower confidence level (LCL) = ?

Upper confidence level (UCL) = ?

a) The number of cities that had only a professional sports team can be found by subtracting the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had all three activities from the total number of cities:

33 - (9 + 6 + 3) = 15 cities had only a professional sports team.

b) The number of cities that had a professional sports team and a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the number of cities that had a professional sports team and a symphony:

9 - 3 = 6 cities had a professional sports team and a symphony, but not a children's museum.

c) The number of cities that had a professional sports team or a symphony can be found by adding the number of cities that had a professional sports team, the number of cities that had a symphony, and then subtracting the number of cities that had both:

17 + 15 - 9 + 14 - 6 + 3 = 34 cities had a professional sports team or a symphony.

d) The number of cities that had a professional sports team or a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the answer to part c:

34 - 3 = 31 cities had a professional sports team or a symphony, but not a children's museum.

e) The number of cities that had exactly two of the activities can be found by adding up the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had a symphony and a children's museum, and then subtracting twice the number of cities that had all three activities:

9 + 6 + 6 - 2(3) = 15 cities had exactly two of the activities.

The volume of the composite figure is 96.3 cm³

First we need to find the volume of the cylinder, and then remove the volume of the rectangular prism.

The radius of the prism is 3cm and the height is 5cm, then the volume is:

V = 3.14*R²*H

V = 3.14*(3cm)²*5cm

V = 141.3 cm³

And the volume of the prism is:

V' = 3cm*3cm*5cm = 45 cm³

The difference gives:

volume = 141.3 cm³ -  45 cm³

volume = 96.3 cm³

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R = 1/8 and the interval of convergence is (-1/8, 1/8).

We can use the ratio test to determine the radius of convergence:

lim_n→∞ |(ε_n+1 - 3)(8x - 3)^n+1 / (ε_n - 3)(8x - 3)^n)|

= lim_n→∞ |(ε_n+1 - 3)/(ε_n - 3)| |8x - 3|

Since the limit of the ratios of consecutive terms is independent of x, we can evaluate it at any particular value of x, such as x = 0:

lim_n→∞ |(ε_n+1 - 3)/(ε_n - 3)| = 1/8

Therefore, the series converges absolutely for |8x - 3| < 1/8, and diverges for |8x - 3| > 1/8. We also need to check the endpoints of the interval:

When 8x - 3 = 1/8, the series becomes Σε_n, which diverges since ε_n is not a null sequence.

When 8x - 3 = -1/8, the series becomes Σ(-1)^nε_n, which converges by the alternating series test, since ε_n is decreasing and approaches zero.

Thus, the interval of convergence is (-1/8, 1/8).

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Options A and D both give an area of 20 square inches

To find the correct dimensions of the parallelogram with an area of 20 square inches, you can use the formula for the area of a parallelogram: Area = base * height.

Given the options:

A. base = 3.06 in, height = 6.54 in
B. base = 6.22 in, height = 3.23 in
C. base = 2.57 in, height = 7.78 in
D. base = 4.00 in, height = 5.00 in

Check each option by plugging the base and height into the formula:

A. 3.06 * 6.54 ≈ 20.00
B. 6.22 * 3.23 ≈ 20.08
C. 2.57 * 7.78 ≈ 19.98
D. 4.00 * 5.00 = 20.00

Options A and D both give an area of 20 square inches. Since the question asks for dimensions to the nearest hundredth of an inch, option A (base = 3.06 in, height = 6.54 in) is more precise and is the correct answer.

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The percent of change on the item 15%

the percent of change in price of the item can be expressed as:

percent of change = ( | new value - old value | / old value) × 100%

Where the vertical bars indicate absolute value.

Given that; the old price was $80 and the new price is $68. So, we can plug these values into the formula:

percent of change = ( | new value - old value | / old value) × 100%

percent of change = ( | 68 - 80 | / 80) × 100%

percent of change = ( | -12 | / 80) × 100%

percent of change = ( 12 / 80) × 100%

percent of change = ( 0.15 ) × 100%

percent of change = 15%

Therefore, Amazon reduced the price of the item by 15%.

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The width of the gravel pathway is 7 meters.

The length of the rectangular garden is 50m and the width is 13m. Let's assume the width of the gravel pathway to be w meters.

The length of the rectangular garden including the two widths of the pathway would be 50+2w meters, and the width including the two widths of the pathway would be 13+2w meters.

The area of the rectangular garden including the pathway is the product of the length and the width:

(50+2w)(13+2w)

We can now set up an equation using the area of the garden and the amount of gravel available:

(50+2w)(13+2w) - 50*13 = 80

Simplifying this equation gives:

4w^2 + 126w - 3196 = 0

This is a quadratic equation that we can solve for w using the quadratic formula:

w = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 4, b = 126, and c = -3196.

Plugging in these values and solving for w gives:

w = 7 or w = -22.75

Since the width of the pathway cannot be negative, the only valid solution is w = 7.

Therefore, the width of the gravel pathway is 7 meters.

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The expression 8(4 - π) yd² is the area of the of the shaded region in terms of π.

The area of the shaded region is the area of the semicircle subtracted from the area of the rectangle

radius of the semicircle is also the width of the rectangle, so;

area of the rectangle = 8 yd × 4 yd = 32 yd²

area of the semicircle = (π × 4 yd × 4 yd)/2

area of the semicircle = 8π yd²

area of the shaded region = 32 yd² - 8π yd²

area of the shaded region = 8(4 - π) yd²

Therefore, the expression 8(4 - π) yd² is the area of the of the shaded region in terms of π.

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The height of the cone is 3.4cm

A cone is a shape formed by using a set of line segments or the lines which connects a common point, called the apex or vertex.

The volume of a cone is expressed as;

V = 1/3 πr²h

where πr² = base area. therefore the volume can be written as;

V = 1/3 base area × height

base area = 49cm²

height = 45.3 cm³

45 = 1/3 ×40h

135 = 40h

h = 135/40

h = 3.4cm

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Answer:

We know that Miranda ran 4 miles in 28 minutes. Let's set up the proportion:

4 miles / 28 minutes = x miles / 2 minutes

To solve for x, we can cross-multiply:

28 minutes * x miles = 4 miles * 2 minutes

28x = 8

Now, let's solve for x by dividing both sides of the equation by 28:

x = 8 / 28

x = 0.2857

Therefore, Miranda runs approximately 0.2857 miles in 2 minutes.

Answer is: manager can be 95% confident that the true proportion of time spent mowing is between 0.452 and 0.548

We can use the normal approximation to the binomial distribution to estimate the confidence interval for the true proportion of time spent mowing.

The sample proportion of time spent mowing is:

p = 200/400 = 0.5

The standard error of the sample proportion is:

SE = sqrt(p(1-p)/n) = sqrt(0.5 * 0.5 / 400) = 0.025

To find the 95% confidence interval for the true proportion of time spent mowing, we can use the formula:

p ± z*SE

where z* is the critical value from the standard normal distribution corresponding to a 95% confidence level, which is approximately 1.96.

So the 95% confidence interval for the true proportion of time spent mowing is:

0.5 ± 1.96*0.025 = (0.452, 0.548)

Therefore, the manager can be 95% confident that the true proportion of time spent mowing is between 0.452 and 0.548.

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