Chapter 4.4, Problem 76E

(a)

To determine

To find: The reason why $a^3-b^3$ is equal to the sum of the volume of the solids I, II and III.

Explanation of Solution

Given information:

Length of the cut-out part=breadth of the cut out part=height of the cut out part $=b$

Length of the whole solid=breadth of the whole solid= height of the whole solid $=a$

Formula used:

Volume = length × breadth × height

  

As we can see from the figure volume of the cut out solid $=b^3$

Volume of the whole solid $=a^3$

Volume of solid, $V(I)=a^2(a-b)$

Volume of solid, $V(II)=(a-b)ab$

Volume of solid, $V(III)=b^2(a-b)$

If we add all these we get $a^3-b^3$

So, from the figure it can be observed that $a^3-b^3$ is equal to the sum of the volume of the solids I, II and III

(b)

To determine

To find: An algebraic expression for the volume of each of the three solids.

Explanation of Solution

Given information:

Length of the cut-out part=breadth of the cut out part=height of the cut out part $=b$

Length of the whole solid=breadth of the whole solid= height of the whole solid $=a$

Formula used:

Volume = length × breadth × height

  

As we can see from the figure volume of the cut out solid $=b^3$

Volume of the whole solid $=a^3$

Volume of solid, $V(I)=a^2(a-b)$

Volume of solid, $V(II)=(a-b)ab$

Volume of solid, $V(III)=b^2(a-b)$

(c)

To determine

To calculate: The results from a) and b) to derive the factoring patterns

Explanation of Solution

Given information:

Length of the cut-out part=breadth of the cut out part=height of the cut out part $=b$

Length of the whole solid=breadth of the whole solid= height of the whole solid $=a$

Formula used:

Volume = length × breadth × height

Calculation:

  

As we can see from the figure volume of the cut out solid $=b^3$

Volume of the whole solid $=a^3$

Volume of solid, $V(I)=a^2(a-b)$

Volume of solid, $V(II)=(a-b)ab$

Volume of solid, $V(III)=b^2(a-b)$

  $V=a^3-a^{2}b+a^{2}b-a{b}^2+a{b}^2-b^3=a^3-b^3$