Question

A closed top box with a square base is to have a volume of 250 cubic meters. The material for the top of the box costs $4 per square meter, the bottom of the box costs $1 per square meter, and the material for the sides costs $6 per square meter. Write an expression that represents the cost of constructing the box in term of x, the side length of the base.

Answer 1

The expression that represents the cost of constructing the box in term of x, the side length of the base is $5 x^{2}+\frac{6000}{x}$

Solution:

Given, a closed top box with a square base is to have a volume of 250 cubic meters.  

The material for the top of the box costs $4 per square meter

The bottom of the box costs $1 per square meter

The material for the sides costs $6 per square meter.  

We have to write an expression that represents the cost of constructing the box in term of x, the side length of the base.

Now, total cost = cost for base + cost for top + cost for sides.

$=1 \times \text { area of base }+4 \times \text { area of top }+6 \times 4 \text { sides } \times \text { area of one side. }$

Assume side of base and top are equal

$=1 \times x^{2}+4 \times x^{2}+6 \times 4 \times x \times \text { height of box }$

$=x^{2}+4 x^{2}+24 \times x \times \frac{\text {volume of box}}{\text {area of base}}$

$=5 x^{2}+24 x \times \frac{250}{x^{2}}=5 x^{2}+24 \times \frac{250}{x}$

$=5 x^{2}+\frac{6000}{x}$

Hence, the expression for cost of box is $5 x^{2}+\frac{6000}{x}$