Question

2 Construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

An open box with a square base is to have a volume of 500 cubic inches. Find the dimensions of the box that will have

minimum surface area. Let x = length of the side of the base.

Show your work:

Answer 1

Answer:

Dimension of box:-

Side of square base = 10 in

Height of box = 5 in

Minimum Surface area, S = 300 in²

Step-by-step explanation:

An open box with a square base is to have a volume of 500 cubic inches.

Let side of the base be x and height of the box is y

Volume of box = area of base × height

                 $500=x^2y$

Therefore, $y=\dfrac{500}{x^2}$

It is open box. The surface area of box, S .

$S=x^2+4xy$

Put  $y=\dfrac{500}{x^2}$

$S(x)=x^2+\dfrac{2000}{x}$

This would be rational function of surface area.

For maximum/minimum to differentiate S(x)

$S'(x)=2x-\dfrac{2000}{x^2}$

For critical point, S'(x)=0

$2x-\dfrac{2000}{x^2}=0$

$x^3=1000$

$x=10$

Put x = 10 into $y=\dfrac{500}{x^2}$

y = 5

Double derivative of S(x)

$S''(x)=2+\dfrac{4000}{x^3}$ at x = 10

$S''(10) > 0$

Therefore, Surface is minimum at x = 10 inches

Minimum Surface area, S = 300 in²