Question

The pyramid of teti has a square base with edge length of 202.5 yards and a slant height of 175.1 yards. Find the height of the pyramid. Round your answer to the nearest tenth.

Answer 1

To find the height of the pyramid, we can use the Pythagorean theorem. The pyramid has a slant height \( l = 175.1 \) yards, and since the base is a square, we can find the distance from the center of the base to the midpoint of one of the edges. This distance is half of the base edge length.

1. The edge length of the base is given as \( 202.5 \) yards.
2. The distance from the center of the base to the midpoint of a side (half of the base length) is:
\[
\frac{202.5}{2} = 101.25 \text{ yards}
\]

Now we have a right triangle where:
- One leg is the height of the pyramid \( h \),
- The other leg is \( 101.25 \) yards (distance from the center to the midpoint of a side),
- The hypotenuse is the slant height \( l = 175.1 \) yards.

Using the Pythagorean theorem:
\[
l^2 = h^2 + (101.25)^2
\]

Plugging in the values:
\[
(175.1)^2 = h^2 + (101.25)^2
\]

Calculating \( (175.1)^2 \):
\[
175.1^2 = 30644.01
\]

Calculating \( (101.25)^2 \):
\[
101.25^2 = 10253.5625
\]

Now substituting back into the equation:
\[
30644.01 = h^2 + 10253.5625
\]

Subtract \( 10253.5625 \) from both sides:
\[
h^2 = 30644.01 - 10253.5625
\]
\[
h^2 = 20390.4475
\]

Now take the square root of both sides to find \( h \):
\[
h = \sqrt{20390.4475} \approx 142.0
\]

Finally, rounding to the nearest tenth, the height of the pyramid is:
\[
\boxed{142.0} \text{ yards}
\]