Question

An open rectangular box with volume 2m^3 has a square base. Express the surface area of the box as a functi...? An open rectangular box with volume 2m^3 has a square base. Express the surface area of the box as a function of the length of a side of the base. I'm not sure how to go about doing this, and I know it sounds silly but please let your reply be easy to read.

Answer 1

The volume of any prism is equal to the area of the base times its vertical height. Since the area of the rectangular prism is a square, the volume is expressed as:

V = s²h

The surface area is equal to the total areas of the faces of the planes. This includes the two bases on top and on the bottom, and the 4 rectangular lateral faces. The rectangular lateral face has an area of its length equal to height h multiplied with the width equal to the side of the square base. So, the surface area is expressed as:

SA = 2s² + 4sh

The first time is twice the area of the base, and the second term is four times the area of the lateral face. So, we want to express the surface area only in terms of s. Therefore, let's substitute an expression in terms of s to the h term above. Let's use the given volume equal to 2 cm³.

V = s²h = 2
Express  in terms of s:
h = 2/s²

Then, let's substitute this to the equation for SA:

SA = 2s² + 4s(2/s²)
SA = 2s² + 8/s