Help with calculus word problem please?

Question

A swimming pool w/ a rectangular surface of 1200ft^2 is to have a cement border area that is 12 ft wide at each end and 8 ft wide at the sides. Find the surface dimensions of the pool if the total area covered is to be a minimum.

Answer 1

Side length of pool (x)
Other side of pool (y)

We know that xy=1200 hence x=1200/y

Add 12 feet to each end of pool and 8 feet to each side

Total length = x+2x12 = x+24
Total width = y+2x8 = y+16

Total Area = (x+24) x (y+16)

Substitute x=1200/y

Total Area = (1200/y+24) x (y+16)
Total Area = 1200 + (16*1200)/y + 24y+24*16
Total Area = 24y + 1584 +19200/y

Differentiate and set equal to zero (for Min)

24-19200/(y^2)=0
24*y^2 - 19200 = 0
24*y^2 = 19200
y^2 = 19200/24 = 800

y=sqrt(800) = 28.28
x = 1200/y = 1200/28.28 = 42.43

Pool dimensions are 28.28 feet by 42.43 feet

Answer 2

It does help to draw a picture, which is a bit difficult to do here, Think of 2 rectangles, one within the other. The outer one is the pool with borders and the inner one is the pool proper. The pool proper has dimension w x l such that w x l = 1200. The pool with borders has dimensions
W = ppw + 24 and L= ppl + 16. We want to minimize the product W x L.

To substitute, W = w + 24 and L = (1200/w )+16.
W x L is then 1200+ 16w + 28800/w + (16)(24).
d (WL)/dw = 16 - 28800/ (w^2). At the minimum,
16 = 28800/(w^2) and w= 42.42 ft. So l=28.28 ft.
With the skirt, W=66.42 ft and l=44.28 ft.