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

$V=[B-b]*[B+b]*h\ units^{3}$