To determine the height of a pyramid given the volume and the side length of its square base, let's follow these steps:
1. Identify the given values:
- Volume [tex]\( V \)[/tex] of the pyramid: [tex]\( 960 \, \text{in}^3 \)[/tex]
- Side length of the square base: [tex]\( 12 \, \text{in} \)[/tex]
2. Calculate the area of the square base:
- The area of a square is found by squaring the side length. Thus, the base area [tex]\( A \)[/tex] can be calculated as:
[tex]\[
A = \text{side length}^2 = 12^2 = 144 \, \text{in}^2
\][/tex]
3. Use the formula for the volume of 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]
- Substitute the known values into the formula:
[tex]\[
960 = \frac{1}{3} \times 144 \times \text{height}
\][/tex]
4. Solve for the height:
- Rearrange the equation to isolate the height:
[tex]\[
960 = \frac{1}{3} \times 144 \times \text{height}
\][/tex]
First, simplify the product of the base area and [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
960 = 48 \times \text{height}
\][/tex]
- Divide both sides by 48 to solve for the height:
[tex]\[
\text{height} = \frac{960}{48} = 20 \, \text{in}
\][/tex]
Therefore, the height of the pyramid is [tex]\( 20 \)[/tex] inches.
