To find the volume of a pyramid with a square base, you can use the formula for the volume of a pyramid:
\[ V = \frac{1}{3} \times (\text{base area}) \times (\text{height}) \]
Given:
- The perimeter of the square base \( P = 18.3 \) inches
- The height of the pyramid \( h = 15.7 \) inches
First, we need to determine the length of one side (\( s \)) of the square base of the pyramid. Since the base is a square, and the perimeter is the sum of all four sides, each side of the square is one fourth the perimeter:
\[ s = \frac{P}{4} \]
Now let's calculate the side length:
\[ s = \frac{18.3 \text{ in}}{4} = 4.575 \text{ in} \]
Next, we calculate the base area (\( A \)) of the square:
\[ A = s^2 \]
\[ A = (4.575 \text{ in})^2 \]
Square the side length:
\[ A = 20.940625 \text{ in}^2 \]
Now we have the base area, so we can use the pyramid volume formula. Plug in the base area and the pyramid height:
\[ V = \frac{1}{3} \times 20.940625 \text{ in}^2 \times 15.7 \text{ in} \]
\[ V = \frac{1}{3} \times 328.7678125 \text{ in}^3 \]
\[ V = 109.5892708333333333 \text{ in}^3 \]
Round the result to the nearest tenth of a cubic inch:
\[ V \approx 109.6 \text{ in}^3 \]
Therefore, the volume of the pyramid, rounded to the nearest tenth, is approximately \( 109.6 \) cubic inches.
