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Step-by-step explanation:

Part C: (Explanation with steps)

Let the length of the swimming pool = l feet

And width of the pool = w feet

Perimeter of the pool = 2(l + w) feet

Since, perimeter of the pool = 8 feet

2(l + w) = 80

l + w = 40

l = (80 - w) -------(1)

Area of the pool = Length × width

A(w) = l × w

By substituting the value of l from equation (1)

A(w) = (80 - w) × w

A(w) = 80w - w²

To find the maximum area of the pool we will find the derivative of the function with respect to 'w' and equate it to zero.

A'(w) = 80 - 2w

(80 - 2w) = 0

w =

w = 40 feet

Therefore, for width (w) = 40 feet area of the pool will be maximum.

From equation (1),

l = 80 - 40

l = 40 feet

Therefore, maximum area of the pool = l × w

                                                               = 40 × 40

                                                               = 1600 square feet

Part A:

Function representing area of the pool,

A(w) = -w² + 80w

Part B:

Maximum area of the surface of the pool = 1600 square feet

step-by-step explanation: