Question Video: Finding the Side Length of a Cube in a Real World Context | Nagwa Question Video: Finding the Side Length of a Cube in a Real World Context | Nagwa

# Question Video: Finding the Side Length of a Cube in a Real World Context

An engineering company makes some aluminum cubes for a customer. Each cube has a surface area of π₯ cmΒ² and a volume of π₯ cmΒ³. What is the length of the side of each cube?

02:27

### Video Transcript

An engineering company makes some aluminum cubes for a customer. Each cube has a surface area of π₯ square centimetres and a volume of π₯ cubic centimetres. What is the length of the side of each cube?

Weβre just going to begin by defining the side length of the cube. Weβre going to let π or π centimetres be the side length of each cube. Weβre told that the surface area of each cube is π₯ square centimetres. Now, of course, the surface area is the combined area of each of the faces. Each of the faces is a square, and so it has an area of π times π, which is π squared. A cube has six faces, so the total surface area of each cube will be six π squared. But of course, we were told that the surface area was π₯, so we can say that π₯ must be equal to six π squared.

Next, weβre told that the volume of each cube is π₯ cubic centimetres. Now, of course, the volume of a cuboid is the length multiplied by the base multiplied by the width. Now, since a cube is simply a cuboid whose lengths are all the same, we can say that the volume of our cube must be π cubed. We were told that the volume is π₯, so we can say that π₯ is equal to π cubed. Since weβve said that π₯ is equal to six π squared but also equal to π cubed, this must mean that six π squared equals π cubed. And now, we have an equation in terms of π that we can solve.

This is a cubic equation, so there could be up to three solutions. So, what weβre not going to do is divide through by π squared. If we do that, weβre going to lose two solutions. Instead, weβre going to do what we do with a quadratic. Weβre going to make equal to zero. So, weβre going to subtract six π squared from both sides. And so, zero is equal to π cubed minus six π squared. We then factor the expression on the right-hand side. Now, each term has a common factor of π squared. So, we get π squared times π minus six. And for π squared times π minus six to be equal to zero, either π squared itself must be equal to zero or π minus six must be equal to zero. We solve this first equation by square rooting both sides. And that gives us π is equal to zero.

Now, of course, we can disregard this. π is the length of a cube and the length of the cube cannot be zero. To solve the second equation, weβll add six to both sides. And that gives us π is equal to six. And so, weβre done. The side length of each cube is six centimetres. Now, in fact, itβs always sensible to check our answers. We check by substituting π equals six into the expression six π squared and π cubed and checking we get the same answer. Six π squared becomes six times six squared, which is 216. π cubed is six cubed, which is also 216. We were hoping that these would be equal and they are, so we know what weβve done is correct. The side length of each cube is six centimetres.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions