Question
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.
To find the length of the edge of the square base, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
In this case, the slant height (c) of the pyramid is the hypotenuse, and the height (a) and the length of the edge of the base (b) are the other two sides of the right triangle.
Using the Pythagorean theorem, we can write the equation as:
c^2 = a^2 + b^2
Substituting the given values:
436.5^2 = 321^2 + b^2
Simplifying:
190583.25 = 103041 + b^2
Rearranging the equation:
b^2 = 190583.25 - 103041
b^2 = 87542.25
Taking the square root of both sides to solve for b:
b = √87542.25
b ≈ 295.9 feet
Therefore, the length of the edge of the square base is approximately 295.9 feet (rounded to the nearest tenth).
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, the height of the pyramid (321 ft) is perpendicular to the base, and the slant height (436.5 ft) is the hypotenuse of a right triangle formed by the height, the edge of the base, and half of one of the diagonals of the base.
Let's call the length of the edge of the square base "l." The other two sides of the right triangle are the height (321 ft) and half the diagonal of the base.
To find the length of the diagonal of the base, we can use the Pythagorean theorem again. We know that the diagonal of a square base divides it into two congruent right triangles.
Using the Pythagorean theorem, we have:
Diagonal^2 = side^2 + side^2
Diagonal^2 = 2 * side^2
Diagonal = sqrt(2) * side
Since the slant height of the pyramid is the hypotenuse of one of the right triangles, we can find the length of the diagonal of the base by dividing the slant height by sqrt(2):
Diagonal = 436.5 ft / sqrt(2)
Now, we have all the information needed to apply the Pythagorean theorem to find the length of the edge of the square base:
l^2 = Diagonal^2 - height^2
l^2 = (436.5 ft / sqrt(2))^2 - (321 ft)^2
l^2 = (436.5 ft)^2 / 2 - (321 ft)^2
l^2 = 190,522.25 ft^2 / 2 - 103,041 ft^2
l^2 = 95,261.125 ft^2 - 103,041 ft^2
l^2 = -7,779.875 ft^2
Since we cannot have a negative length, it seems there might be an error or inconsistency in the given information or calculations. Please double-check the values provided or verify if there are any additional details that could help resolve the discrepancy.
nope step by step bot?
Apologies for the error in the previous response. Here's a revised step-by-step solution:
1. We have the height of the pyramid as 321 feet (labeled in the image) and the slant height as 436.5 feet. Our goal is to find the length of the edge of the square base, denoted as "l."
2. Recognize that the height, slant height, and length of the base edge form a right triangle within the pyramid.
3. Use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
4. Apply the Pythagorean theorem to the right triangle formed by the height, the base edge length, and half of one of the diagonals of the base.
l^2 + (321^2 / 2)^2 = 436.5^2
5. Simplify the equation:
l^2 + (102241 / 2)^2 = 190622.25
6. Compute 102241 / 2:
l^2 + 51120.5^2 = 190622.25
7. Square 51120.5:
l^2 + 2612771041 = 190622.25
8. Subtract 2612771041 from both sides:
l^2 = 190622.25 - 2612771041
l^2 = -2612570418.75
9. Since we cannot have a negative length, it appears that there is an error or inconsistency in the given information or calculations. Please double-check the values provided or verify if there are any additional details that could help resolve the discrepancy.
To find the length of the edge of the square base, we can use the Pythagorean theorem.
In a square pyramid, the slant height, height, and length of the edge of the base form a right triangle.
Let's denote the length of the edge of the base as "l."
Using the Pythagorean theorem, we can write:
l^2 + (321)^2 = (436.5)^2
l^2 + 103041 = 190622.25
l^2 = 190622.25 - 103041
l^2 = 87581.25
Taking the square root of both sides, we get:
l = √87581.25
l ≈ 296.1
Therefore, the length of the edge of the square base is approximately 296.1 feet.