Answer to Question #274598 in Calculus for Artika

Question

A container with square base, vertical sides, and open top is to have a volume of 1000 . Find the dimensions of the container with minimum surface area.

Answer 1

Solution;

Let the base be of dimension x by x while the height is y.

The constraint equation is ;

$V=x^2y=1000cm^3$

The objective equation is of the S.A;

$S.A=4xy+x^2$

Using constraint equation;

$y=\frac{1000}{x^2}$

Substitute into the objective equation;

$S.A=\frac{4000}{x}+x^2$

For minimum S.A,set derivative equal to zero;

$S.A'=-\frac{4000}{x^2}+2x=0$

$2x^3=4000$

$x^3=2000$

$x=12.6cm$

$y=\frac{1000}{12.6^2}=6.3cm$