at a movie theater, the size of popcorn bags decreased 20% . if the old bags held 15 cups of popcorn, how much do the new bags hold?

Answer:

10:9:1

Step-by-step explanation:

Final answer:

The side length of the largest cube that fits between two concentric spheres with radii of 15 and 33 is approximately 17.32.

Explanation:

To find the side length of the largest cube that fits between two concentric spheres (with the same center), we need to understand that the cube will fit within the smaller sphere and will be inscribed within it such that the diagonals of the cube's faces will be equal to the diameter of the smaller sphere.

The radius of the smaller sphere is given as 15.

Therefore, the diameter of this sphere, which is twice the radius, is 30.

Using the diagonal of a cube, which can be calculated with the formula √3 times the side length (s), it is found that

√3 × s = 30.

Solving for s gives us the maximum side length of the cube that would fit:

s = 30 / √3

s ≈ 17.32

Therefore, the side length of the largest cube is approximately 17.32.

Answer:

A binomial probability density function should be used because the question asks for determining the probability that exactly 10 students wear glasses.

Step-by-step explanation:

A binomial probability density function is used when we want to know the probability of an exact value.

A cumulative distribution function is used when we want to know the probability of less than or equal to a value.

Answer:

256 boys.

Step-by-step explanation:

Let x represent number of boys in the course.

We have been given that there are 416 students who are enrolled in an introductory Chinese course. There are eight boys to every five girls.

We will use proportions to find the total number of boys in the course.

Since there are eight boys to every five girls, so there are 8 boys for 13 (8+5) total students.

[tex]\frac{\text{Boys}}{\text{Total students}}=\frac{8}{13}[/tex]

Upon substituting our values in above equation, we will get:

[tex]\frac{x}{416}=\frac{8}{13}[/tex]

Multiply both sides by 416:

[tex]\frac{x}{416}*416=\frac{8}{13}*416[/tex]

[tex]x=8*32[/tex]

[tex]x=256[/tex]

Therefore, there are 256 boys in the course.

Final answer:

To calculate the number of boys in a class with a given ratio of boys to girls, divide the total number of students by the sum of the parts of the ratio to find one part's value, then multiply by the number of parts for boys. In this case, with 416 students and a ratio of 8 boys to 5 girls, there are 256 boys enrolled in the course.

Explanation:

Calculating the Number of Boys and Girls in a Class

To find the number of boys in an introductory Chinese course with a ratio of eight boys to every five girls, we can use the concept of ratios and proportions. Given that there are 416 students enrolled in the course, we first add the parts of the ratio together, which gives us 8 (boys) + 5 (girls) = 13 parts in total. The number of boys can then be calculated as follows:

Divide the total number of students by the total number of parts to find the value of one part: 416 students ÷ 13 parts = 32 students per part.

Multiply the value of one part by the number of parts attributed to the boys: 32 students per part × 8 parts (for boys) = 256 boys.

Therefore, based on the ratio provided, there are 256 boys enrolled in the introductory Chinese course.

Answer:

Here is a more clearer version of the same question ;

29 1.2. Constructing Direct Proofs 1. Let a, b, and c be real numbers with a 0. The solutions of the quadratic equation ax 2 + bx + c = 0 are given by the quadratic formula, which states that the solutions are xi and x2. where and x2 2a 2a (a) Prove that the sum of the two solutions of the quadratic equation ax2 + bx + c = 0 is equal to -b/a . (b) Prove that the product of the two solutions of the quadratic equation ax2 + bx + c = 0 is equal to c/a.

The proving of both a) and b) has been done

Step-by-step explanation:

The step by step explanation has been given in the attachment below.

Answer:

ff

Step-by-step explanation:

Answer:

[tex]25t+19\leq 144[/tex]

[tex]t\leq5[/tex]

The number of minutes she can continue to descend if she does not want to reach a point more than 144 feet below the ocean surface is at most 5 minutes.

Step-by-step explanation:

Given:

Initial depth of the scuba dive = 19 ft

Rate of descent = 25 ft/min

Maximum depth to be reached = 144 ft

Now, after 't' minutes, the depth reached by the scuba dive is equal to the sum of the initial depth and the depth covered in 't' minutes moving at the given rate.

Framing in equation form, we get:

Total depth = Initial Depth + Rate of descent × Time

Total depth = [tex]19+25t[/tex]

Now, as per question, the total depth should not be more than 144 feet. So,

[tex]\textrm{Total depth}\leq 144\ ft\\\\19+25t\leq 144\\\\or\ 25t+19\leq 144[/tex]

Solving the above inequality for time 't', we get:

[tex]25t+19\leq 144\\\\25t\leq 144-19\\\\25t\leq 125\\\\t\leq \frac{125}{25}\\\\t\leq 5\ min[/tex]

Therefore, the number of minutes she can continue to descend if she does not want to reach a point more than 144 feet below the ocean surface is at most 5 minutes.

Answer: The answer is D

Step-by-step explanation:

Since the first digit must not be zero, and the first three digits must not be 900 or 800, the highest ten digits telephone number we can get is 899, since it is the highest number that is next in line to the forbidden numbers

The number of possible 10-digit telephone numbers, considering the given restrictions, is 638,000.

To find the number of possible 10-digit telephone numbers, we need to consider the restrictions given. The first digit cannot be zero, so there are 9 options for the first digit. The next 2 digits cannot be 800 or 900, so there are 8 options for each of these digits. The remaining 6 digits can be any number from 0 to 9, so there are 10 options for each of these digits.

Therefore, the total number of possible telephone numbers is 9 x 8 x 8 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 648,000. However, the number must end in 0000, so we subtract the cases where the last 4 digits are not all zeros. There are 10 options for each of these 4 digits, so there are 10 x 10 x 10 x 10 = 10,000 cases where the number does not end in 0000.

Therefore, the final number of possible telephone numbers is 648,000 - 10,000 = 638,000. The correct answer is B) 638,000.

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The hockey player must score 20 goals in order to make the same money as the first contract

Solution:

Given that, hockey player is offered two options for a contract

Let "x" be the number of goals made

A base salary of 50,000 and 1000 per goal

Therefore, money earned is given as:

Money earned = 50000 + 1000(number of goals)

Money earned = 50000 + 1000x ------- eqn 1

Base salary of 40,000 and 1500 per goal

Therefore, money earned is given as:

Money earned = 40000 + 1500(number of goals)

Money earned = 40000 + 1500x --------- eqn 2

In order to make the same money as the other contract, eqn 1 must be equal to eqn 2

50000 + 1000x = 40000 + 1500x

1500x - 1000x = 50000 - 40000

500x = 10000  

x = 20

Thus he must score 20 goals in order to make the same money as the first contract

Answer:

The answer to your question is Long = 10.39

Step-by-step explanation:

Data

hypotenuse = 12

long = ?

Process

1.- To find Long, we must use the trigonometric functions sine or cosine.

If we use sine, we use the 60° angle

If we use cosine, we use the 30° angle

a) sin 60 = long / hypotenuse

   Long = hypotenuse x sin 60

   Long = 12 x sin 60

   Long = 10.39

b) cos 30 = Long / hypotenuse

   Long = hypotenuse x cos 30

   Long = 12 x cos 30

  Long = 10.39

Answer:

Step-by-step explanation:

We have the triangle 30° - 60° - 90°. The sides are in ratio 1 : 2 : √3.

(short : hypotenuse : long)

We have hypotenuse = 12.

Therefore

short = 12/2 = 6

long = 6√3

Answer:

Assuming he works at a constant rate, the daily rate Mark rakes lawns in the fall is 40% of the daily rate he mows lawns in the summer.

Step-by-step explanation:

Fall:

5 neighbours' lawns / 2.5 hours = 2 lawns per hour

Summer:

6 neighbours' lawns / 1.2 hours = 5 lawns per hour

2 x 100% / 5 = 40%

2 is 40% of 5

The percent of the daily rate should be 40%.

One day in the fall, Mark raked the leaves on 5 neighbors’ lawns in 2.5 hr. One day in the summer, he mowed 6 neighbors’ lawns in 1.2 hr.

For fall

= 5 neighbours' lawns ÷  2.5 hours = 2 lawns per hour

For Summer:

= 6 neighbours' lawns ÷ 1.2 hours = 5 lawns per hour

Now

[tex]= 2 \times 100\% \div 5[/tex]= 40%

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Answer:

  4

Step-by-step explanation:

Put the numbers in the expression in place of the corresponding variables and do the arithmetic.

  [tex]\dfrac{8^2}{2^4}=\dfrac{64}{16}=4[/tex]

Answer:

4

Step-by-step explanation:

Answer: Using Pythagoras Rule the ladder touches 13.75ft ~ 14ft up the wall.

Step-by-step explanation: Using the Pythagoras Rule the Hypothenus is the leaning hieght of the ladder 15ft

And the base is 6ft. Now let's find the height of the wall h¹

h¹ = √{15² - 6²}

h¹ =√{225 - 36}

h¹ = √189

= 13.74ft

Answer:

A

Step-by-step explanation:

(3n²+2n+4)(2n-1)

to solve this multiply through by using each term in one bracket to multiply all terms in the second bracket to open the bracket.

3n²(2n-1) + 2n(2n-1) +4(2n-1)

6n³-3n² +4n²-2n+8n-4

6n³+n²+6n-4

Hence the correct option is A

Answer:

a

Step-by-step explanation:

Answer:

The rider is 15.91  seconds  50 ft above the ground after reaching the low point

Step-by-step explanation:

We can evaluate the angle  α

using trigonometry applied to the orange small triangle with height 50-30 = 20 ft and  hypotenuse equal to the radius  r = 25ft

Now

[tex]20 = \frac{25}{sin(\alpha)}[/tex]

[tex]\alpha = arcsin{\frac{20}{25}[/tex]

[tex]\alpha = 53.13^{\circ}[/tex]

So  50 ft  of height corresponds to the total angle:

[tex]90^{\circ} =53.13^{\circ} = 143.13 6^{\circ}[/tex]  = 2.498 radians

Now the angular velocity

[tex]\omega = \frac{2\pi}{T}[/tex]

[tex]\omega = \frac{2 \pi}{40}[/tex]

[tex]\omega =0.157rad/s[/tex]

To describe  2.498  rad  it will take:

[tex]t = \frac{2.498}{0.157}[/tex]

t = 15.91 s

The rider reaches 50 feet above the ground approximately 4.09 seconds after reaching the lowest point.

To solve the problem of determining how long after reaching the low point a rider on the Ferris wheel is 50 feet above the ground, we can follow these steps:

Step 1: Define the parameters

Diameter of the Ferris wheel: 50 feet.

Radius of the Ferris wheel: [tex]\( r = \frac{50}{2} = 25 \)[/tex] feet.

Center of the wheel height: 30 feet above the ground.

Rotation period: 40 seconds per revolution.

Step 2: Establish the equation for the height of a rider

The vertical position ( h(t) ) of a rider at time ( t ) seconds can be modeled using the cosine function (assuming the lowest point at ( t = 0 ):

[tex]\[ h(t) = 30 + 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]

Step 3: Set the height equation to 50 feet

Since we want the height h(t) to be 50 feet:

[tex]\[ 50 = 30 + 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]

Step 4: Solve for the cosine term

[tex]\[ 50 = 30 + 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]

[tex]\[ 20 = 25 \cos\left(\frac{2\pi t}{40}\right) \][/tex]

[tex]\[ \cos\left(\frac{2\pi t}{40}\right) = \frac{20}{25} \][/tex]

[tex]\[ \cos\left(\frac{2\pi t}{40}\right) = 0.8 \][/tex]

Step 5: Determine the angle

[tex]\[ \frac{2\pi t}{40} = \cos^{-1}(0.8) \][/tex]

[tex]\[ \frac{2\pi t}{40} = 0.6435 \][/tex]

[tex]\[ t = \frac{0.6435 \times 40}{2\pi} \][/tex]

[tex]\[ t \approx 4.09 \text{ seconds} \][/tex]

Thus, the rider reaches 50 feet above the ground approximately 4.09 seconds after reaching the lowest point.

Answer:

The scale factor is 2

Step-by-step explanation:

Scale factor  [tex]k=\frac{Image\: length}{Corresponding\:Object\:Length}[/tex]

[tex]k=\frac{|A'B'|}{|AB|}[/tex]

We can use the Pythagoras Theorem to get:

[tex]k=\frac{\sqrt{4^2+4^2} }{\sqrt{2^2+2^2}}[/tex]

[tex]k=\frac{\sqrt{2*4^2} }{\sqrt{2*2^2}}=\frac{4\sqrt{2} }{2\sqrt{2} } =2[/tex]

The scale factor is 2

Answer:

(1,-5)

Step-by-step explanation:

recall that for any function g(x), we can equate the function to y,

i.e y = g(x), where any point (x,y) that satisfies the equation is considered a point on the graph

in our case we are given, that g(1) = -5, rearranging:

-5 = g(1)

if we compare this with the the equation above y = g(x),

we can see clearly that y = -5 and x = 1.

and from our reasoning given in the paragraph above, we can say that (1,-5) is a point on the graph.

solving 1/3h+5 is greater then or equal to 1/6h+1 ([tex]\frac{1}{3}h+5\geq \frac{1}{6}h+1[/tex]) we get [tex]h\geq -24[/tex]

Step-by-step explanation:

We need to solve 1/3h+5 is greater then or equal to 1/6h+1

Writing in mathematical form:

[tex]\frac{1}{3}h+5\geq \frac{1}{6}h+1[/tex]

Solving and finding value of h

Adding -5 on both sides

[tex]\frac{1}{3}h+5-5\geq \frac{1}{6}h+1-5\\\frac{1}{3}h\geq \frac{1}{6}h-4[/tex]

Adding -1/6h on both sides

[tex]\frac{1}{3}h-\frac{1}{6}h\geq \frac{1}{6}h-4-\frac{1}{6}h\\\frac{1*2h-1*1h}{6}\geq -4\\\frac{2h-1h}{6}\geq -4\\\frac{1h}{6}\geq -4\\h\geq -4*6\\h\geq -24[/tex]

So, solving 1/3h+5 is greater then or equal to 1/6h+1 ([tex]\frac{1}{3}h+5\geq \frac{1}{6}h+1[/tex]) we get [tex]h\geq -24[/tex]

Keywords: Solving inequalities

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Answer: E. Parametric estimating

Step-by-step explanation:                          

A parametric estimate is the estimating process that uses a statistical relationship between past project's data and other variables to give an estimation for the parameters such as duration ,cost and budget.

It is based on parameters that summarize the risk , costs of algorithm , project , process , complexity and service.

It can create a greater level of accuracy.

Therefore , Parametric estimating is an estimating technique that uses a statistical relationship between historical data and other variables to calculate an estimate for activity parameters such as duration and cost.

Hence , the correct answer is E. Parametric estimating.

Answer:

9 yards

Step-by-step explanation:

Let the length of each quilt be 'l'.

Given:

Length of green felt (g) = 15 yards

Length of blue felt (b) = 12 yards

Total number of quilts = 3

Now, the length of 3 quilts is given as:

[tex]Total\ length=l+l+l=3l[/tex] ----- (1)

As per question:

Total length = Length of green felt + Length of blue felt

Total length = [tex]g+b[/tex]

[tex]Total\ length=15+12=27[/tex]--------- (2)

Now, comparing equations (1) and (2), we get:

[tex]3l=27\\\\l=\frac{27}{3}\\\\l=9\ yards[/tex]

Hence, each quilt measures 9 yards in length.

Answer:

The signal will not be picked up if he drives from the first city to the second city as far as he drives along a straight line.

Step-by-step explanation:

The attachment to the answer clearly explains why no signal will be picked up

Answer:

The answer is 38.5 miles

Step-by-step explanation:

From the attached diagram

AC represents the distance needed to be traveled before picking the signal.

Using Pythagoras'rule to solve for AC from triangle ACO

(AO)^2 = (AC)^2 + (CO)^2

(60)^2 = (AC)^ + (46)^2

(AC) = ✓[(60)^2 - (46)^2]

AC = 38.5 miles

Answer:

  8

Step-by-step explanation:

We assume there are 2 bushings per bolt, and 2 bolts per spring. Whether you count springs and multiply by 2, or count bushings and divide by 2, you get 8 eyebolts.

_____

Comment on the question

There are conceivable mechanical arrangements in which the number of eyebolts might be some other number. There is not actually enough information here to properly answer the question.

To replace all the bushings in a Jeep YJ, a total of 8 leaf spring eyebolts would be required considering each spring uses two eyebolts.

To be able to replace all the bushings in your Jeep YJ's leaf springs, you need to first establish how many leaf spring eyebolts you will need. In a standard leaf spring setup, each end of the spring (both front and back) is held in place by a leaf spring eyebolt. Since you have 4 springs and each spring uses 2 leaf spring eyebolts (one for the front leaf spring eye and one for the back), you'll require a total of 8 leaf spring eyebolts for your Jeep YJ.

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Answer:

  22 1/2 hours

Step-by-step explanation:

There are several ways you can get there. One way is to use table values to form a sum of 60:

  cars = 32×2 - 8×(1/2) = 64 -4 = 60

Then the corresponding hours are ...

  hours = 12×2 - 3×(1/2) = 24 -1.5 = 22.5

It will take 22.5 hours to wax 60 cars.

__

Another way to get there is to use proportions:

  hours/cars = 3/8 = ??/60

Multiply by 60 to find ??

  (60)(3/8) = ?? = 180/8 = 22.5 . . . . hours

_____

We can do these things because we have determined the relationship between hours and cars is proportional: 8/3 = 16/6 = 32/12. If it weren't proportional, some other strategy would be needed to answer the question.

The coordinates of point B that could be correct are (-2, 2) and (-3, 7). (option b and f)

Given that line AB is perpendicular to line BC, the product of their slopes will be -1. This means that the slope of line AB will be the negative reciprocal of the slope of line BC.

The slope of line BC can be calculated as

(7 - 2) / (2 - 2) = 5 / 0.

However, the denominator being zero implies that the slope is undefined, which contradicts the basic definition of a slope. This means that line BC is vertical, and its slope cannot be calculated.

Since line BC is vertical, line AB must be horizontal, and its slope is 0. Therefore, for a horizontal line passing through point A (-3, 2), any value of y will be 2. This means that the y-coordinate of point B must also be 2.

Let's analyze the options:

A. (8, 0): Incorrect y-coordinate.

B. (-2, 2): Correct y-coordinate.

C. (-4, 5): Incorrect y-coordinate.

D. (1, 3): Incorrect y-coordinate.

E. (-1, -1): Incorrect y-coordinate.

F. (-3, 7): Correct y-coordinate.

From the analysis, the correct coordinates for point B are (-2, 2) and (-3, 7).

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Answer:

D. 21

Step-by-step explanation:

Given:

m arc AC = 69

Segment AB is tangent to circle O at point A.

We need to find the ∠ ABC

Solution:

Now we can say that;

By inscribed angle theorem which states that;

"Central angle is equal to arc subtended by it."

m∠O = 69 (Central angle)

Also Given:

Segment AB is tangent to circle O at point A.

Now by radius tangent property which states that;

"Radius which touches to closet point on the tangent is always perpendicular."

so we can say that;

m∠OAB = 90

Now in Δ OAB.

m∠O = 69

m∠OAB = 90

Now we know that;

"Sum of all angles of triangle is 180."

m∠O + m∠OAB + m∠ABO = 180

Substituting the values we get;

[tex]69+90+m\angle ABO=180\\\\159+m\angle ABO=180[/tex]

Subtracting both side by 159 we get;

[tex]159+m\angle ABO-159=180-159\\\\m\angle ABO=21[/tex]

Now we can say that;

m∠ABO = m∠ ABC = 21 (same angles)

Hence  m∠ ABC is 21.

Answer: the number of four point questions in the test is 10.

the number of five point questions in the test 8

Step-by-step explanation:

Let x represent the number of four point questions in the test.

Let y represent the number of five point questions in the test.

There are 18 questions on the test. It means that

x + y = 18

Each question is worth either four or five points. The total is 80 points. This means that

4x + 5y = 80 - - - - - - - - - - - - - -1

Substituting x = 18 - y into equation 1, it becomes

4(18 - y) + 5y = 80

72 - 4y + 5y = 80

- 4y + 5y = 80 - 72

y = 8

x = 18 - y = 18 - 8

x = 10

Answer:

b

Step-by-step explanation:

for sure

Answer:

  h = f(t) = -15cos((π/5)t) +20

Step-by-step explanation:

If you like, you can make a little table of positions:

  (t, h) = (0, 5), (5, 35)

Since the wheel is at an extreme position at t=0, a cosine function is an appropriate model:

  h = Acos(kt) +C

The amplitude A of the function is half the difference between the t=0 value and the t=5 value:

  A = (1/2)(5 -35) = -15

The midline value C is the average of the maximum and minimum:

  C = (1/2)(5 + 35) = 20

The factor k satisfies the relation ...

  k = 2π/period = 2π/10 = π/5

So, the function can be written as ...

  h = f(t) = -15cos((π/5)t) +20

The equation of the function of the height in meters above the ground t minutes after the wheel begins to turn, is presented as follows;

[tex]\mathbf{h = f(t) = 15 \cdot sin\left[ \left(\dfrac{\pi}{5} \right) \cdot (t - 2.5)\right] + 20}[/tex]

The process for finding the above function is as follows;

The given parameters of the Ferris wheel are;

The diameter of the Ferris wheel, D = 30 meters

The height of the Ferris wheel above the ground, d = 5 meters

The level of loading the platform = The six o'clock position

The time it takes the wheel to make one full revolution, T = 10 minutes

The required parameter;

The function which gives the height, h, of the Ferris wheel = f(t)

Method;

The equation that can be used to model the height of the Ferris wheel is the sine function which is presented as follows;

y = A·sin[k·(t - b)] + c

Where;

A = The amplitude = (highest point - Lowest point)/2

∴ A = (35 - 5)/2 = 15

The period, T = 2·π/k

∴ 10 minutes = 2·π/k

k = 2·π/10 = π/5

k = π/5

At the starting point, the Ferris wheel is at the lowest point, and t = 0, we have;

The sine function is at the lowest point when k·(t - b) = -π/2

Therefore, we get;

π/5·(0 - b) = -π/2

-b = -π/2 × (5/π) = -5/2 = -2.5

b = 2.5

The vertical shift, c = (Max - Min)/2 + Min = (Max + Min)/2

∴ c = (35 + 5)/2 = 20

c = 20

The equation of the Ferris wheel of the form, y = A·sin[k·(t - b)] + c, is therefore;

[tex]\mathbf{h = f(t) = 15 \cdot sin\left[ \left(\dfrac{\pi}{5} \right) \cdot (t - 2.5)\right] + 20}[/tex]

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Answer:

Karla needs to bur 4 sandwiches.

Step-by-step explanation:

Given:

Amount of subway sandwich each person would get = [tex]\frac{1}{2}[/tex]

Number of people coming to watch football game = 7

Now given:

She wants to make sure each person,including herself,will get 1/2 of a subway sandwich.

Total number of people would watch the match = 8 persons

We need to find the number of sandwiches she need to buy.

Solution:

Now we know that;

to find the number of sandwiches she need to buy can be calculated by multiplying Amount of subway sandwich each person would get with Total number of people would watch the match.

framing in equation form we get;

the number of sandwiches she need to buy = [tex]\frac{1}{2}\times8 = 4[/tex]

Hence Karla needs to bur 4 sandwiches.

Answer:

  -4 ≤ y < ∞

Step-by-step explanation:

The graph shows the range (vertical extent) has a minimum value of -4, and includes all y-values greater than or equal to that value.

In interval notation, the range is [-4, ∞).

The dimensions of lake are length 1.5 miles and width 0.3 miles

Solution:

Given that,

Artificial lake is in the shape of a rectangle

Let the length of lake be "a"

The width of the lake is 1/5 the length of the lake

[tex]width = \frac{length}{5}\\\\width = \frac{a}{5}[/tex]

The area of lake is 9/20 square miles

The area of rectangle is given by formula:

[tex]Area = length \times width[/tex]

Substituting the values we get,

[tex]\frac{9}{20} = a \times \frac{a}{5}\\\\a^2 = \frac{9}{20} \times 5\\\\a^2 = \frac{9}{4}\\\\\text{Taking square root on both sides }\\\\a = \frac{3}{2} = 1.5[/tex]

Thus, we get

[tex]length = a = 1.5 \text{ miles }\\\\width = \frac{a}{5} = \frac{1.5}{5} = 0.3 \text{ miles }[/tex]

Thus the dimensions of lake are length 1.5 miles and width 0.3 miles