Question

the top of the prism can be decomposed to find the volume of the empty space outside the 2 given cubes.

1. determine the dimensions and volume of prism 1 and prism 2. a list of possible dimensions is provided.

	possible dimensions
prism	dimensions			volume
1	__, , and __			○ b²(a - b) ○ ab(a - b)
2	__, , and __			○ b²(a - b) ○ ab(a - b)

1. complete the steps to find the factored form of a³ + b³.

step	expression
2	factor out (a - b) from both expressions.	(a - b)(____)
3	factor out the gcf from the second binomial.	__(a - b)(__)
4	subtract the combined volume of prisms 1 and 2 from the volume of the entire rectangular prism found in problem 1.	a²(a + b) - b(a - b)(a + b)
5	factor out (a + b) from both expressions.	(a + b)(____)
6	rewrite the expression after distribution.	(a + b)(____)

Explanation:

Step1: Determine dimensions and volumes of prisms

For Prism 1, dimensions are \(a\), \(a + b\), and \(a - b\). Volume \(V_1=a(a + b)(a - b)=a(a^{2}-b^{2})=a^{3}-ab^{2}\). But looking at the options, if we consider the base - height concept, if the base has area \(a(a - b)\) and height \(b\), the volume \(V_1 = ab(a - b)\). Dimensions are \(a\), \(b\), and \((a - b)\).
For Prism 2, dimensions are \(b\), \(b\), and \((a - b)\). Volume \(V_2=b\times b\times(a - b)=b^{2}(a - b)\).

Step2: Combine volumes of prisms

The combined volume of Prism 1 and Prism 2 is \(V = ab(a - b)+b^{2}(a - b)\).

Step3: Factor out \((a - b)\)

\(V=(a - b)(ab + b^{2})\).

Step4: Factor out GCF from second binomial

Factor out \(b\) from \((ab + b^{2})\), we get \(V = b(a - b)(a + b)\).

Step5: Subtract combined volume from total volume

We are given the expression for subtraction \(a^{2}(a + b)-b(a - b)(a + b)\).

Step6: Factor out \((a + b)\)

\(a^{2}(a + b)-b(a - b)(a + b)=(a + b)(a^{2}-b(a - b))=(a + b)(a^{2}-ab + b^{2})\).

Step7: Recall the identity for \(a^{3}+b^{3}\)

We know that \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\).

Answer:

Prism 1: \(a\), \(b\), \((a - b)\), Volume: \(ab(a - b)\)
Prism 2: \(b\), \(b\), \((a - b)\), Volume: \(b^{2}(a - b)\)
Step 1: \(ab(a - b)+b^{2}(a - b)\)
Step 2: \((a - b)(ab + b^{2})\)
Step 3: \(b(a - b)(a + b)\)
Step 5: \((a + b)(a^{2}-ab + b^{2})\)
Step 6: \((a + b)(a^{2}-ab + b^{2})\)