Question

A rectangular box is to have a square base and a volume of 5 ft3 . The material for the base costs 10 cents/ft2 , the material for the top costs 30 cents/ft2 , and the material for the sides costs 50 cents/ft2 . If x denotes the length of one side of the base, find a function in the variable x giving the total cost of materials in cents.

Answer 1

Answer:

$f(x)=40x^2+\frac{100}{x}$

Step-by-step explanation:

Let $x$ denote side of the square base.

Area base=Area top:

$Area=side\times side\\=x^2$

#Area of base and top is $x^2$ each

Given the volume as 5 cubic feet and side of base as x, height is:

$Volume=x^2h\\5=x^2h\\h=5/x^2\\\\\therefore:\\Area \sides=4\times 5/x^2 \times x\\\\A_s=20/x$

Substitute the given prices  to give the function for cost of material.

Top=30cents/ft, Base=10cents/ft and sides =50cents/ft:

$f(x)=10x^2+30x^2+50(20/x)\\\\f(x)=10x^2+30x^2+\frac{100}{x}\\\\f(x)=40x^2+\frac{100}{x}$

The function for the cost of material is expressed as:$f(x)=40x^2+\frac{100}{x}$