A stainless steel patio heater is a square pyramid. The length of one side of the base is 22.2 in. The slant height of the pyramid is 90.1 in. What is the height of the pyramid?
The Correct Answer and Explanation is:
To find the height of the pyramid, we need to use the Pythagorean Theorem, as the slant height, height, and half the base form a right triangle. Here’s how you can calculate it:
Step 1: Understanding the components
- Base of the pyramid: The base is square, and the side length of the square is given as 22.2 inches.
- Slant height: The slant height (denoted as lll) is 90.1 inches. This is the distance from the apex of the pyramid to the midpoint of a side of the base along the face of the pyramid.
- Height: This is the perpendicular distance from the apex of the pyramid to the center of the base (what we need to find).
Step 2: Apply the Pythagorean Theorem
The slant height, the height of the pyramid, and half the side length of the base form a right triangle, with the slant height as the hypotenuse.
- The half of the side length of the base is 22.22=11.1\frac{22.2}{2} = 11.1222.2=11.1 inches. This is one leg of the right triangle.
- The height of the pyramid (denoted as hhh) is the other leg.
- The slant height (denoted as lll) is the hypotenuse, which is 90.1 inches.
Using the Pythagorean theorem:l2=(side length of the base2)2+h2l^2 = \left(\frac{\text{side length of the base}}{2}\right)^2 + h^2l2=(2side length of the base)2+h2
Substitute the values:90.12=11.12+h290.1^2 = 11.1^2 + h^290.12=11.12+h28101.01=123.21+h28101.01 = 123.21 + h^28101.01=123.21+h2h2=8101.01−123.21h^2 = 8101.01 – 123.21h2=8101.01−123.21h2=7977.80h^2 = 7977.80h2=7977.80h=7977.80h = \sqrt{7977.80}h=7977.80h≈89.37 inchesh \approx 89.37 \text{ inches}h≈89.37 inches
Step 3: Conclusion
The height of the pyramid is approximately 89.37 inches.
This calculation uses the Pythagorean theorem, where the slant height is the hypotenuse, and half the base length and the height of the pyramid form the two other sides of the right triangle.
