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 costs $2.50 per square foot; and I spent $450 altogether on material for the shed.
Express the volume of the shed as a function of the (length of each) side of the square base.

Answer 1

Answer:

$V = \frac{450L - 6 L^{3} }{10}$

Step-by-step explanation:

The volume of a cuboid can be determined simply by the formula: V= LWH

(where: L is length, H is height and W is width).

In this particular case the base is a square, which means the length and width are equal. Hence we can modify the equation of volume:

$V = L^{2} H$

Now we need to find the value of H in terms of L. For this we can develop the equation for cost incurred in building the storage shed. We find the area of each side of the cuboid, and then we multiply it by cost per square feet to find the total cost incurred; as shown below:

Area:

Base: $L$×$L = L^{2}$

Roof: $L$×$L = L^{2}$

Side: $4 H$×$L = 4HL$  (we have considered all four sides)

Cost:

Base: 4$L^{2}$

Roof: 2$L^{2}$

Side: $2.50 * 4HL = 10 HL$

Total cost:

4$L^{2}$ + 2$L^{2}$ + 10 $HL$ = 450

We simplify this equation further:

6$L^2$ + 10HL = 450

10HL = 450 - 6$L^2$

$H = \frac{450- 6 L^2}{10L}$  

We now have the value of H, which we can substitute in the formula of Volume we deduced earlier:

substituting $H = \frac{450-6L}{10L}$  in  $V = L^{2} H$ :

$V = L^2$ × $\frac{450 - 6L }{10L}$

Simplifying it further:

$V = L$ × $\frac{450 - 6L}{10}$

$V= \frac{450L - 6L^3}{10}$ is the final answer.