Extrema swimming pools Rectangle center,half circle at one end. Which will have

Question

Extrema swimming pools

Rectangle center,half circle at one end. Which will have a max area with perimeter of 100 ft.
Find the first derivative and first derivative equal to zero (that is, if you graphed the crazy function for area it would have a horizontal tangent line sitting at the top of the curve and the input value underneath would be the critical point).
Provide dimensions and area of pool in your sketches. Check that your final dimensions comply with perimeter of 100 feet.
Verify maximum with second derivative test.

Explanation / Answer

pool with shape Rectangle center,half circle at one end:

Let dia of semi circle be x = width of pool

Let length of pool be y

Perimeter : pi(x/2) + pi(x/2) + 2x +2y = 100

x( pi +2) + 2y = 100

y = 50 - x(pi +2)/2

Area = x*y = x(50 - x/2(pi +2))

= 50x - (x^2/2)(pi +2)

dA/dx = 50 - x(pi+2) ; dA/dx =0 ; x = 50/(pi+2)

d^2A/dx^2 = -(pi+2) ; -ve only maxima occurs

x = 9.72 ft ; y = 50 - 9.72(pi +2)/2 = 25 ft

Dimensions of Pool : x= 9.72 ft ; y = 25 ft

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