(Solved): Dr. Summer wants to design a square concrete-lined pool having a box-like shape (see diagram below) ...

Dr. Summer wants to design a square concrete-lined pool having a box-like shape (see diagram below) and a volume of \( 500 \mathrm{ft}^{3} \). She wants the (concrete-lined) surface area to be as small as possible. Note that she doesn't have the top covered because it is swimming season. She finds that the surface area of the pool can be represented by the function \( A(x)=x^{2}+\frac{2000}{x} \), where \( x \) is the width of the pool. Find the width, length, and height \( (y) \) that will minimize the surface area of the pool, assuming Dr. Summer wants the width to be at least \( 4 f t \) but no more than \( 20 f t \).