Question

An open-top box with a square base is to have a volume of 4 cubic feet. Find the dimensions of the box (in ft) that can be made with the smallest amount of material.

Answer 1

Answer:

The dimension of box=$2ft\times 2ft\times 1ft$

Step-by-step explanation:

We are given that

Volume of box=4 cubic feet

Let x be the side of square base and h be the height of box

Volume of box=$lbh=x^2h$

$4=x^2h$

$h=\frac{4}{x^2}$

Now, surface area of box,A=$x^2+4xh$

$A=x^2+4x(\frac{4}{x^2})=x^2+\frac{16}{x}$

$\frac{dA}{dx}=2x-\frac{16}{x^2}$

$\frac{dA}{dx}=0$

$2x-\frac{16}{x^2}=0$

$2x=\frac{16}{x^2}$

$x^3=8$

$x=2$

$\frac{d^2A}{dx^2}=2+\frac{32}{x^3}$

Substitute x=2

$\frac{d^2A}{dx^2}=2+\frac{32}{2^3}=2+4=6>0$

Hence, the area of box is minimum at x=2

Therefore, side of square base,x=2 ft

Height of box,h=$\frac{4}{2^2}=1 ft$

Hence, the dimension of box=$2ft\times 2ft\times 1ft$