Question

A crate open at the top has vertical sides, a square bottom, and a volume of 500 cubic meters. Using the labels shown in the figure, write the surface area S as a function of x.

A) S(x) = 4x²
B) S(x) = 6x²
C) S(x) = 8x²
D) S(x) = 10x²

Answer 1

To write the surface area S as a function of x for a crate with a volume of 500 cubic meters, we can determine the area of each side and sum them up. The total surface area is given by S(x) = x^2 + 4x(500/x^2), which simplifies to S(x) = x^2 + 2000/x.

Step-by-step explanation:

To write the surface area S as a function of x, we need to determine how each side of the crate contributes to the total surface area. The bottom of the crate is a square with side length x, so its area is x^2. The four vertical sides of the crate have a height of x and a width of 500/x^2 (since the volume is 500 cubic meters). Therefore, the total surface area is:

S(x) = x^2 + 4x(500/x^2)

Simplifying the expression:

S(x) = x^2 + 2000/x