The pyramid is an arena in Memphis, Tennessee. It's height is 321 ft., and it's slant height is 436.5 ft. Find the length of the edge of the square base. Round the answer to the nearest tenth.
To find the length of the edge of the square base of the pyramid, we can use the Pythagorean theorem. Since the slant height goes from the midpoint of one of the sides of the square base to the top of the pyramid, we can consider a right triangle formed by the slant height (the hypotenuse of the right triangle), half the length of the base (one of the legs of the triangle), and the height of the pyramid (the other leg of the triangle).
Let's denote:
- \( s \) as the slant height (given as 436.5 ft)
- \( h \) as the height of the pyramid (given as 321 ft)
- \( x \) as half the length of the base
Using the Pythagorean theorem:
\[ s^2 = h^2 + x^2 \]
Plugging in the values we have:
\[ (436.5)^2 = (321)^2 + x^2 \]
Now let's solve for \( x \):
\[ 190632.25 = 103041 + x^2 \]
\[ x^2 = 190632.25 - 103041 \]
\[ x^2 = 87591.25 \]
\[ x = \sqrt{87591.25} \]
\[ x \approx 295.96 \]
Now, remember, \( x \) is only half the length of the base, so to find the full length of the edge of the square base, we multiply \( x \) by 2:
\[ \text{Edge length of the base} = 2 \times x \]
\[ \text{Edge length of the base} \approx 2 \times 295.96 \]
\[ \text{Edge length of the base} \approx 591.92 \]
Rounded to the nearest tenth, the edge length of the square base is approximately 591.9 feet.
Incorrect
Let's correct the calculation. If the height of the pyramid is 321 feet and the slant height is 436.5 feet, we are dealing with a right-angled triangle where the pyramid's height is one leg of the triangle, half the diagonal of the base is the other leg, and the slant height is the hypotenuse.
Denote the following:
- \( s \) as the slant height, which is 436.5 feet.
- \( h \) as the height of the pyramid, which is 321 feet.
- \( d/2 \) as half the diagonal length of the square base.
The square base's diagonal can be expressed in terms of the edge length of the base. If \( e \) is the edge length of the base, then the diagonal \( d \) is given by \( d = e\sqrt{2} \) based on the properties of a square.
Using the Pythagorean theorem:
\[ s^2 = h^2 + (d/2)^2 \]
Plug in the values we have:
\[ 436.5^2 = 321^2 + (e\sqrt{2}/2)^2 \]
\[ 190562.25 = 103041 + (e^2 \cdot 2)/4 \]
We want to solve for \( e^2 \):
\[ 4 \cdot 190562.25 = 4 \cdot 103041 + e^2 \cdot 2 \]
\[ 762249 = 412164 + 2e^2 \]
Subtract \( 412164 \) from both sides:
\[ 350085 = 2e^2 \]
Divide by 2:
\[ e^2 = \frac{350085}{2} \]
\[ e^2 = 175042.5 \]
Take the square root of both sides:
\[ e = \sqrt{175042.5} \]
\[ e \approx 418.38 \]
Rounded to the nearest tenth, the edge length of the square base is approximately 418.4 feet.
To find the length of the edge of the square base of the pyramid, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the slant height of the pyramid acts as the hypotenuse, and the height of the pyramid and the length of the edge of the square base are the other two sides of the right triangle.
Let's assume the length of the edge of the square base is x. Therefore, the height of the pyramid is also x.
Using the Pythagorean theorem, we have:
x^2 + x^2 = 436.5^2
Simplifying the equation:
2x^2 = 436.5^2
Now, solve for x by dividing both sides by 2:
x^2 = (436.5^2) / 2
x ≈ 308.3 (rounded to the nearest tenth)
Therefore, the length of the edge of the square base of the pyramid is approximately 308.3 feet.