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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.

Answer 1

To find the length of the edge $l$ of the square base of the pyramid given its height $h=321$ feet and its slant height $s=436.5$ feet, we can use the relationship between the height, the slant height, and half the base length.

When we draw a vertical line from the apex of the pyramid down to the center of the base, this line (height $h$) creates a right triangle with the slant height $s$ and half the base edge length $\frac{l}{2}$ as the other two sides.

Using the Pythagorean theorem in this right triangle, we can write the equation:

$$s^2=h^2+{\left(\frac{l}{2}\right)}^2$$

Substituting the known values:

$$(436.5{)}^2=(321{)}^2+{\left(\frac{l}{2}\right)}^2$$

Calculating the squares:

$$190,504.25=103,041+{\left(\frac{l}{2}\right)}^2$$

Now, subtract $103,041$ from both sides:

$$190,504.25-103,041={\left(\frac{l}{2}\right)}^2$$

$$87,463.25={\left(\frac{l}{2}\right)}^2$$

Taking the square root of both sides to solve for $\frac{l}{2}$:

$$\frac{l}{2}=\sqrt{87,463.25}\approx 295.7$$

Now, multiplying by 2 to get the side length $l$:

$$l\approx 2\times 295.7\approx 591.4$$

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

$$\boxed{{\displaystyle 591.4\text{ feet}}}$$