Answer to Question #136455 in Calculus for Gab

Question

A rectangular open box is have a square base, and its volume is to be 125 inches^3. The cost Per in^2 Of the material for the bottom Is 8$ and for the sides is 4$.

A. Find the mathematical model Expressing The total cost Of the material As a function of the edge Lenght Of the square base.

B. What is the total cost If the square Base has edge Lenght of 4 inches

Answer 1

A. Denote length of the side of the square base by $x.$ The volume $V$ of a rectangular box is equal to $x^2h$, where $h$ is the heigth of the box. Therefore, $h=\frac{V}{x^2}=\frac{125}{x^2}$. The area of the lateral sides is equal to $4\frac{125}{x^2}x=\frac{500}x$. Thus, the mathematical model expressing the total cost of the material can be written in the following form:

$m(x)=8x^2+4\frac{500}{x}=8x^2+\frac{2000}{x}$

B. If the square base has edge length of $x=4$ inches, then its total cost is

$m(4)=8\cdot 4^2+\frac{2000}{4}=128+500=628.$