Answer to Question #342171 in Calculus for Amy Lee

Question

A small jewelry box with square of base is to have a volume of 125 cu.cm. Find its dimensions to require the least amount of material.

Answer 1

The Volume of a box with a square base x by x cm and height h cm is $V=x^2h$

The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.

The surface area of the box described is $A=x^2+4xh$  

We need A as a function of x alone, so we'll use the fact that $V=x^2h=125cm^3$

which gives us $h=\frac{V}{x^2}=\frac{125}{x^2}$ ,so the area becomes:

$A=x^2+4xh=x^2+4x\frac{125}{x^2}=x^2+\frac{500}{x}$

We want to minimize A, so

$A`=2x-\frac{500}{x^2}=0$ when $\frac{2x^3-500}{x^2}=0$

$2x^3-500=0$

$2x^3=500$

$x^3=250$

The only critical number is $x=5\sqrt[3]2$

The second derivative test verifies that A has a minimum at this critical number:

$A``=2+\frac{1000}{x^3}$ which is positive at $x=5\sqrt[3]2$

The box should have base $5\sqrt[3]2\thickapprox 6.29961$ cm by $5\sqrt[3]2\thickapprox 6.29961$ cm and height 3.1498 cm.

$h=\frac{V}{x^2}=\frac{125}{(5\sqrt[3]2)^2}\thickapprox3.1498$