Question

A square pyramid is shown. Its height is shown with a dotted line labeled 321 feet. Its slant height is shown with a dotted line labeled 436.5 feet. The variable l is shown along one of the edges of the square base.
The Pyramid is an arena in Memphis, Tennessee. Its height is 321 ft., and its slant height is 436.5 ft. Find the length of the edge of the square base. Round the answer to the nearest tenth.

(1 point)

Answer 1

To find the length of an edge of the square base of the pyramid, we can use the relationship between the height, slant height, and half the length of the base edge.

Let:
- \( h \) be the height of the pyramid, which is 321 feet.
- \( l \) be the slant height of the pyramid, which is 436.5 feet.
- \( b \) be the length of an edge of the square base.

The relationship we can use here involves the right triangle formed by the height \( h \), half of the base \( \frac{b}{2} \), and the slant height \( l \). According to the Pythagorean theorem:

\[
l^2 = h^2 + \left(\frac{b}{2}\right)^2
\]

We can rearrange this equation to solve for \( b \):

\[
\left(\frac{b}{2}\right)^2 = l^2 - h^2
\]

\[
\frac{b}{2} = \sqrt{l^2 - h^2}
\]

\[
b = 2\sqrt{l^2 - h^2}
\]

Now, plug in the values of \( h \) and \( l \):

1. Calculate \( l^2 \) and \( h^2 \):
- \( l^2 = (436.5)^2 = 190,609.25 \)
- \( h^2 = (321)^2 = 103,041 \)

2. Subtract \( h^2 \) from \( l^2 \):
- \( l^2 - h^2 = 190,609.25 - 103,041 = 87,568.25 \)

3. Take the square root:
- \( \frac{b}{2} = \sqrt{87,568.25} \approx 295.9 \)

4. Multiply by 2 to find \( b \):
- \( b \approx 2 \times 295.9 \approx 591.8 \)

Thus, the length of the edge of the square base, rounded to the nearest tenth, is:

\[
\boxed{591.8} \text{ feet}
\]