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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.

(1 point

Answer 1

To find the length of the edge of the square base of the pyramid, we can use the Pythagorean theorem. In a square pyramid, the height, half of the base edge, and the slant height form a right triangle.

1. Let's denote:

   • Height (h) = 321 ft
   • Slant height (l_s) = 436.5 ft
   • Half of the base edge (a) = x/2, where x is the length of the edge of the base.

2. According to the Pythagorean theorem: $$l_s^2=h^2+a^2$$

   Substituting the known values: $$(436.5{)}^2=(321{)}^2+(x/2{)}^2$$

3. Calculate $(436.5{)}^2$ and $(321{)}^2$: $$(436.5{)}^2=190,505.25$$ $$(321{)}^2=103,041$$

4. Substitute these values back into the equation: $$190,505.25=103,041+(x/2{)}^2$$

5. Rearranging gives: $$(x/2{)}^2=190,505.25-103,041$$ $$(x/2{)}^2=87,464.25$$

6. Taking the square root: $$x/2=\sqrt{87,464.25}\approx 295.7$$

7. Therefore, to find x: $$x=2\times 295.7\approx 591.4$$

Thus, the length of the edge of the square base rounded to the nearest tenth is approximately 591.4 feet.