Answered

The base of a solid right pyramid is a square with an edge length of [tex]n[/tex] units. The height of the pyramid is [tex]n-1[/tex] units.

Which expression represents the volume of the pyramid?

A. [tex]\frac{1}{3} n(n-1) \, \text{units}^3[/tex]
B. [tex]\frac{1}{3} n(n-1)^2 \, \text{units}^3[/tex]
C. [tex]\frac{1}{3} n^2(n-1) \, \text{units}^3[/tex]
D. [tex]\frac{1}{3} n^3(n-1) \, \text{units}^3[/tex]



Answer :

GinnyAnswer
Let's determine the volume of the pyramid step-by-step given the details.

### Step 1: Identify the base area

The base of the pyramid is a square with edge length [tex]\( n \)[/tex] units. The formula for the area of a square is:

[tex]\[ \text{Base Area} = \text{side}^2 = n^2 \][/tex]

### Step 2: Identify the height

The height of the pyramid from the base to the apex is given as [tex]\( n - 1 \)[/tex] units.

### Step 3: Use the formula for the volume of a pyramid

The volume [tex]\( V \)[/tex] of a pyramid with a base area [tex]\( B \)[/tex] and height [tex]\( h \)[/tex] is given by:

[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]

### Step 4: Substitute the base area and height into the volume formula

From the problem:
- Base Area [tex]\( B = n^2 \)[/tex]
- Height [tex]\( h = n - 1 \)[/tex]

Substituting these into the volume formula, we get:

[tex]\[ V = \frac{1}{3} \times n^2 \times (n - 1) \][/tex]
[tex]\[ V = \frac{1}{3} n^2 (n - 1) \][/tex]

### Step 5: Compare with the given options

Let's compare the expression we derived with the given options:

1. [tex]\( \frac{1}{3} n(n-1) \)[/tex] units[tex]\(^3\)[/tex]
2. [tex]\( \frac{1}{3} n(n-1)^2 \)[/tex] units[tex]\(^3\)[/tex]
3. [tex]\( \frac{1}{3} n^2(n-1) \)[/tex] units[tex]\(^3\)[/tex]
4. [tex]\( \frac{1}{3} n^3(n-1) \)[/tex] units[tex]\(^3\)[/tex]

The expression that matches our derived formula is:

[tex]\[ \frac{1}{3} n^2 (n - 1) \][/tex]

### Conclusion

Hence, the correct expression that represents the volume of the pyramid is:

[tex]\[ \boxed{\frac{1}{3} n^2(n-1)} \][/tex] units[tex]\(^3\)[/tex].

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