Choose the one alternative that best completes the statement or answers the question.
-A rectangular box with volume 433 cubic feet is built with a square base and top. The cost is $\$ 1.50$ per square foot for the top and the bottom and $\$ 2.00$ per square foot for the sides. Let $x$ represent the length of a side of the base. Express the cost the box as a function of $x$.
A) $C ( x ) = 3 x ^ { 2 } + \frac { 3,464 } { x }$
B) $C ( x ) = 4 x + \frac { 3,464 } { x ^ { 2 } }$
C) $C ( x ) = 3 x ^ { 2 } + \frac { 1,732 } { x }$
D) $C ( x ) = 2 x ^ { 2 } + \frac { 3,464 } { x }$

Question

Quizplus · Accepted Answer

The cost function $C(x)$ combines the cost of the top and bottom, which have an area of $x^2$ each and cost $1.50 per square foot, with the cost of the sides. The volume of the box is $433$ cubic feet, so the height $h$ can be expressed as $\frac{433}{x^2}$. There are four sides, each with an area of $xh = x\left(\frac{433}{x^2}\right) = \frac{433}{x}$, and each costs $2.00 per square foot. Therefore, the total cost is $C(x) = 2(x^2)(1.5) + 4\left(\frac{433}{x}\right)(2) = 3x^2 + \frac{3464}{x}$.

Choose the One Alternative That Best Completes the Statement or Answers