- Identify the base as a square with area \(B = n^2\).
- Identify the height of the pyramid as \(h = n - 1\).
- Apply the general volume formula for a pyramid: \(V = \frac{1}{3}Bh\).
- Substitute the identified values to obtain the final expression: \(\boxed{\frac{1}{3} n^2(n - 1) \, \text{units}^3}\).
Explanation
- Determine the volume expression for the pyramid.
A right pyramid has a base and an apex. The volume \(V\) of any pyramid is given by the formula:
\[V = \frac{1}{3} \times (\text{Area of the base}) \times (\text{height})\]
In this problem, the base of the pyramid is a square with an edge length of \(n\) units. Therefore, the area of the square base, which we denote as \(B\), is:
\[B = n \times n = n^2\]
The height of the pyramid is given as \(h = n - 1\) units.
Now, we substitute the area of the base and the height into the volume formula:
\[V = \frac{1}{3} \times n^2 \times (n - 1)\]
Thus, the expression representing the volume of the pyramid is \(\frac{1}{3} n^2(n - 1)\).
- Match the expression with the provided options.
Comparing the derived expression \(V = \frac{1}{3} n^2(n - 1)\) with the given options:
- \(\frac{1}{3} n(n-1) \, \text{units}^3\)
- \(\frac{1}{3} n(n-1)^2 \, \text{units}^3\)
- \(\frac{1}{3} n^2(n-1) \, \text{units}^3\)
- \(\frac{1}{3} n^3(n-1) \, \text{units}^3\)
The expression \(\frac{1}{3} n^2(n-1) \, \text{units}^3\) is clearly the third option.
Examples
Imagine you are a packaging designer working on a project for a client who wants to create souvenir boxes in the shape of a pyramid. If the square base of these boxes has a side length of \(n = 5\) cm and the pyramid's height is \(n - 1 = 4\) cm, you can easily calculate the required volume for storage capacity using our formula:
\[V = \frac{1}{3} \times 5^2 \times 4 = \frac{1}{3} \times 25 \times 4 = \frac{100}{3} \approx 33.33 \, \text{cm}^3\]
This helps in determining how much content can fit inside the box, allowing for precise material planning and efficient design.