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.

Question

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 of the square base \( l \) of the pyramid, we can use the relationship between the height, the slant height, and half the length of the edge of the base.

1. The height (\( h \)) of the pyramid is given as 321 feet.
2. The slant height (\( s \)) is given as 436.5 feet.
3. The relationship between the height, the slant height, and half the length of the edge of the base can be expressed using the Pythagorean theorem.

The relationship can be defined as: \[ s^2 = h^2 + \left(\frac{l}{2}\right)^2 \]

Substituting the values: \[ (436.5)^2 = (321)^2 + \left(\frac{l}{2}\right)^2 \]

Calculating the squares: \[ 436.5^2 = 190,196.25 \] \[ 321^2 = 103,683 \]

Now, substituting these back into the equation: \[ 190,196.25 = 103,683 + \left(\frac{l}{2}\right)^2 \]

Rearranging the equation to solve for \( \left(\frac{l}{2}\right)^2 \): \[ \left(\frac{l}{2}\right)^2 = 190,196.25 - 103,683 \] \[ \left(\frac{l}{2}\right)^2 = 86,513.25 \]

Taking the square root: \[ \frac{l}{2} = \sqrt{86,513.25} \approx 293.7 \]

Now, multiplying by 2 to find the length \( l \): \[ l \approx 2 \times 293.7 \approx 587.4 \]

Thus, the length of the edge of the square base \( l \), rounded to the nearest tenth, is: \[ \boxed{587.4} \text{ feet} \]

Answer 2

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Answer 3

Let's work through the problem again to ensure we find the correct length of the edge of the square base of the pyramid.

Given:

• Height \( h = 321 \) feet
• Slant height \( s = 436.5 \) feet

We need to find the length of the edge of the square base \( l \).

Using the relationship involving the height, the slant height, and half the length of the edge of the base, we can employ the Pythagorean theorem again:

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

Substituting the known values:

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

Calculating the squares:

1. Calculate \( (436.5)^2 \): \[ 436.5^2 = 190,196.25 \]

2. Calculate \( (321)^2 \): \[ 321^2 = 103,683 \]

Now substituting these values into the equation:

\[ 190,196.25 = 103,683 + \left(\frac{l}{2}\right)^2 \]

Rearranging to find \( \left(\frac{l}{2}\right)^2 \):

\[ \left(\frac{l}{2}\right)^2 = 190,196.25 - 103,683 \] \[ \left(\frac{l}{2}\right)^2 = 86,513.25 \]

Taking the square root:

\[ \frac{l}{2} = \sqrt{86,513.25} \approx 293.7 \]

Now multiplying by 2 to find \( l \):

\[ l \approx 2 \times 293.7 \approx 587.4 \]

The previous computation seems correct upon re-evaluation, which indicates that the length of the edge of the square base, rounded to the nearest tenth, is:

\[ \boxed{587.4} \text{ feet} \]

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