To determine which expression correctly represents the volume of the right pyramid given the parameters, let's analyze the problem in a detailed, step-by-step manner.
Step 1: Understanding the Volume Formula for a Pyramid
The volume [tex]\( V \)[/tex] of a pyramid is given by the formula:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
Step 2: Identifying the Base Area
The base of the pyramid is a square with an edge length of [tex]\( n \)[/tex] units. The area of a square is calculated as:
[tex]\[ \text{Base Area} = n^2 \][/tex]
Step 3: Identifying the Height
The height of the pyramid is given as [tex]\( n-1 \)[/tex] units.
Step 4: Substitute the Base Area and Height into the Volume Formula
Now, substituting the base area and the height into the volume formula, we get:
[tex]\[ V = \frac{1}{3} \times n^2 \times (n-1) \][/tex]
Thus, the expression for the volume of the pyramid becomes:
[tex]\[ \frac{1}{3} n^2 (n-1) \, \text{units}^3 \][/tex]
Step 5: Check the Given Options
1. [tex]\(\frac{1}{3} n (n-1)^2\)[/tex] units[tex]\(^3\)[/tex]
2. [tex]\(\frac{1}{3} n^2 (n-1)\)[/tex] units[tex]\(^3\)[/tex]
3. [tex]\(\frac{1}{3} n^2 (n-1)\)[/tex] unit[tex]\(^3\)[/tex]
Upon examining the options, the correct expression for the volume we derived is:
[tex]\[ \frac{1}{3} n^2 (n-1) \, \text{units}^3 \][/tex]
Notice the units in both options considered as correct formulas for the volume:
- The first option has an incorrect formula.
- The second option has a correct formula and the units correctly denoted as cubic units.
- The third option has an incorrect unit notation (singular), since volume should be denoted in cubic units.
So, the correct answer is:
[tex]\[
\frac{1}{3} n^2 (n-1) \, \text{units}^3
\][/tex]
Conclusion
Given the parameters and calculations, the expression that represents the volume of the pyramid is:
[tex]\[ \frac{1}{3} n^2(n-1) \, \text{units}^3 \][/tex]
