Use the image to answer the 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)
ft.
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To find the length of the edge of the square base of the pyramid, we can use the Pythagorean theorem. The slant height of the pyramid is the hypotenuse of a right triangle formed by the height and half of the length of the base.
Let's call the length of the base l.
Using the Pythagorean theorem, we can write:
l^2 + (l/2)^2 = 436.5^2
Simplifying the equation:
l^2 + l^2/4 = 191000.25
Multiplying both sides by 4 to eliminate the fraction:
4l^2 + l^2 = 764001
Combining like terms:
5l^2 = 764001
Dividing both sides by 5:
l^2 = 152800.2
Taking the square root of both sides:
l ≈ √152800.2
l ≈ 390.6
Therefore, the length of the edge of the square base of the pyramid is approximately 390.6 feet (rounded to the nearest tenth).
In order 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 side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we can consider the slant height (436.5 ft) as the hypotenuse and the height (321 ft) as one of the other sides. The length of the edge of the square base (l) would be the other side.
Using the Pythagorean theorem, we have:
l^2 + 321^2 = 436.5^2
Now we can solve for l:
l^2 + 103041 = 191078.25
l^2 = 88037.25
Taking the square root of both sides, we find:
l ≈ 296.6 ft.
So, the length of the edge of the square base of the pyramid is approximately 296.6 ft.
To find the length of the edge of the square base, we can use the formula for the slant height of a square pyramid:
slant height^2 = height^2 + (1/2 * edge length)^2.
Since we are given the slant height (436.5 ft) and the height (321 ft), we can plug these values into the formula and solve for the edge length.
(436.5 ft)^2 = (321 ft)^2 + (1/2 * edge length)^2
Simplifying this equation gives:
191041.25 ft^2 = 103041 ft^2 + (1/4 * edge length)^2
Now we can isolate the term with the edge length:
(1/4 * edge length)^2 = 191041.25 ft^2 - 103041 ft^2
(1/4 * edge length)^2 = 88000.25 ft^2
Taking the square root of both sides:
(1/4 * edge length) = √88000.25 ft^2
(1/4 * edge length) = 296.6667 ft
Finally, solving for the edge length:
edge length = 296.6667 ft * 4
edge length = 1186.67 ft
Therefore, the length of the edge of the square base is approximately 1186.7 ft.