Question

The Derivative in Graphing and Applications

23. A closed rectangular container with a square base is to have a volume of 2000cm^3. It costs twice as much per square centimeter for the top and bottom as it does for the sides. Find the dimensions of the container of least cost.

PLEASE HELPP!!!

Answer 1

\[cost \propto C(x)=4hx+\color{brown}4x^2\]


\[C(x)={8000\over x}+4x^2\]

at least cost: \(C'(x)=0\)

Answer 2

\[8x-{8000 \over x^2}=0\]\[\implies 8x^3=8000\]\[\implies x=10\]\[\implies h={2000 \over 10^2}=20\]