Question

The figure below shows a container that is a square prism with base side length B and a hollow section (shaded region) that is a square prism with base side length b. If the height h of both prisms is the same, write an expression to represent the volume of the container. Express your answer in completely factored form.

Answer 1

we know that

The volume of a square prism is equal to

`V=A*H`

where

A is the area of the base of the prism

H is the height of the prism

The volume of the container is equal to the volume of the larger prism minus the volume of the smaller prism

Step 1

Find the volume of the larger prism

`V=A*H`

In this case

`A=B^(2)\ units^(2)`

`H=h\ units`

`V=B^(2)*h\ units^(3)`

Step 2

Find the volume of the smaller prism

`V=A*H`

In this case

`A=b^(2)\ units^(2)`

`H=h\ units`

`V=b^(2)*h\ units^(3)`

Step 3

Find the volume of the container

The volume of the container is equal to the volume of the larger prism minus the volume of the smaller prism

so

`V=(B^(2)*h)\ units^(3)-(b^(2)*h)\ units^(3)`

Simplify

`V=[B^(2)-b^(2)]*h\ units^(3)`

Difference of squares

`V=[B-b]*[B+b]*h\ units^(3)`

therefore

the answer is

`V=[B-b]*[B+b]*h\ units^(3)`