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

       The volumes would be calculated as: (the Base area)* (the height) ;
 assuming equal units;
 and the volume would be expressed in:  "units³ " ; or, "cubic units".
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    Note that EACH of these square prisms have a "square" as the base;  and a square has the same length as the width ; 
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So the base area of the square prism with side length, "B", would
    be: "B * B" ; or, "B² " .  The height would be, "h".
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The volume would be expressed as:  V = B² * h  ; or, " V = B² h" .
 
In completely factored form, the volume would be expressed as:
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              V = B * B * h .
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Likewise, the volume for the "shaded" square prism would be:
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       V = b² * h ;  or,  " V = b² h " .
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In "completely factored form", the volume could be expressed as:
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                  V = b * b * h  .
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Answer 2

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

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