A closed box with a square base is to have a volume of 20 cubic meters. The material for the top and bottom of the box costs $3 per square meter, and the material for the sides costs $1 per square meter. Express the construction cost of the box as a function of the length of its base, x.
A) $C ( x ) = x ^ { 2 } + 80 x + 2$ dollars
B) $C ( x ) = 2 x ^ { 2 } + \frac { 2 } { x } + 6$ dollars
C) $C ( x ) = 2 x ^ { 2 } + \frac { 20 } { x }$ dollars
D) $C ( x ) = 6 x ^ { 2 } + \frac { 80 } { x }$ dollars

Question

Quizplus · Accepted Answer

The cost function is derived by considering the cost of the top and bottom, which is $6x^2$ (since both top and bottom cost $3 per square meter), and the sides, which cost $\frac{80}{x}$ (since the volume constraint gives height as $\frac{20}{x^2}$ and the total side area is $4x \times \frac{20}{x^2}$).

A Closed Box with a Square Base Is to Have C(x)=x2+80x+2C ( x ) = x ^ { 2 } + 80 x + 2C(x)=x2+80x+2

A Closed Box with a Square Base Is to Have $C ( x ) = x ^ { 2 } + 80 x + 2$