Question

A circus tent is in the shape of a square pyramid. Each edge of the square base is 20 m and its slant height is 13.5 m. Find the height of the tent. Round the answer to the nearest tenth.(1 point)
Responses

16.8 m 
16.8 m 

9.1 m 
9.1 m 

24.1 cm
24.1 cm

14.8 cm

Answer 1

To find the height of the square pyramid (the tent) with a square base and a given slant height, we can use the Pythagorean theorem.

1. **Identify the dimensions**:
- The length of each edge of the square base (\(b\)) = 20 m.
- The slant height (\(l\)) = 13.5 m.

2. **Find the length of the diagonal from the center of the base to a vertex**:
- The distance from the center of the base to the midpoint of one of the edges is \( \frac{b}{2} = \frac{20 \, \text{m}}{2} = 10 \, \text{m} \).
- The full diagonal from the center of the base to a vertex can be calculated using the Pythagorean theorem:
\[
\text{distance from center to vertex} = \sqrt{\left(\frac{b}{2}\right)^2 + \left(\frac{b}{2}\right)^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2} \, \text{m} \approx 14.14 \, \text{m}
\]

3. **Apply the Pythagorean theorem** to find the height (\(h\)):
- In the right triangle formed by the height \(h\), the slant height \(l\), and the distance from the center to a vertex, we have the equation:
\[
l^2 = h^2 + \left(\frac{b}{2}\right)^2
\]
- Plugging in the values we know:
\[
(13.5)^2 = h^2 + (10)^2
\]
\[
182.25 = h^2 + 100
\]
\[
h^2 = 182.25 - 100
\]
\[
h^2 = 82.25
\]
\[
h = \sqrt{82.25} \approx 9.1 \, \text{m}
\]

Therefore, the height of the circus tent, rounded to the nearest tenth, is **9.1 m**.