Verified answer

Step-by-step explanation:

a. Let's represent the length of the swimming pool as "l" and the width as "w."

b. The equation that represents the perimeter of the swimming pool is: 2l + 2w = 86.

c. The equation that represents the area of the swimming pool is: l * w = 450.

d. To find the length and width, we can solve the system of equations formed by the perimeter and area equations. We have:

2l + 2w = 86 [Equation 1]

l * w = 450 [Equation 2]

Using substitution or elimination, we can solve for l and w. Let's use substitution:

From Equation 1, we get: 2l = 86 - 2w

Simplifying, we have: l = (86 - 2w)/2

Now substitute this expression in Equation 2:

(86 - 2w)/2 * w = 450

(86w - 2w^2)/2 = 450

86w - 2w^2 = 900

2w^2 - 86w + 900 = 0

To solve this quadratic equation, we can factor or use the quadratic formula. Solving it, we find two possible values for w: w = 18 and w = 25.

Substituting these values back into Equation 1, we can find the corresponding lengths.

For w = 18, l = (86 - 2*18)/2 = 25

For w = 25, l = (86 - 2*25)/2 = 18

Therefore, the length of the swimming pool can be 25 m and the width can be 18 m.

e. To check if the dimensions satisfy the given conditions, ensure that the perimeter and area equations hold true when substituting the obtained values. Plug in l = 25 and w = 18 into the equations:

For the perimeter: 2 * 25 + 2 * 18 = 50 + 36 = 86

For the area: 25 * 18 = 450

Both conditions are satisfied, confirming that the dimensions are correct.

f. If the dimensions of the swimming pool are both doubled, the new length would be 50 m (2 * 25) and the new width would be 36 m (2 * 18).

For the perimeter: 2 * 50 + 2 * 36 = 100 + 72 = 172 m

For the area: 50 * 36 = 1800 m^2

Doubling the dimensions increases the perimeter to 172 m and the area to 1800 m².