Question

I built a storage shed in the shape of a rectangular box on a square base. The material that I used for the base cost $4 per square foot, the material for the roof cost $2 per square foot, and the material for the sides cost $2.50 per square foot; and I spent $450 altogether on material for the shed. What should the side of the base be in order to maximize the volume of the shed?

Answer 1

Answer:

side of 6.124 ft and height of 3.674 ft

Step-by-step explanation:

Let's s be the side of the square base and h be the height of the rectangular box.

The base and the roof would have an area of $s^2$ and cost of

$4s^2 + 2s^2 = 6s^2$

The sides would have an area of 4sh and cost of 4sh*2.5 = 10sh

So the total cost for the material is

$6s^2 + 10sh = 450$

$10sh = 450 - 6s^2$

$h = 45/s - 0.6s$

The volume of the shed has the following formula

$V = s^2h = s^2(45/s - 0.6s) = 45s - 0.6s^3$

To find the maximum value for V, we can take its first derivative, and set it to 0

$\frac{dV}{ds} = 45 - 1.2s^2 = 0$

$1.2s^2 = 45$

$s^2 = 45 / 1.2 = 37.5$

$s = \sqrt{37.5} = 6.124 ft$

h = 45/s - 0.6s = 3.674 ft