### Problem 12: Surface Area of a Right Pyramid (Excluding the Base)**Problem Statement:**Imagine a right pyramid with a square base (similar to an Egyptian pyramid). Suppose that the sides of the square base are all 200 yards (yd) long and that the distance from each vertex of the base to the apex of the pyramid (measured along an edge) is 245 yards (yd). Determine the surface area of the pyramid, excluding the area of the base. Explain your reasoning.**Solution:**To find the surface area of the pyramid excluding the base, we need to calculate the area of the four triangular faces.1. **Calculate the Slant Height (l):** The slant height is the height of each triangular face from the middle of one side of the base to the apex of the pyramid. Each side of the square base is 200 yards. - Half of one side of the base = \( \frac{200}{2} = 100 \: \text{yd} \) - Using Pythagoras' theorem in the right triangle with base half of the side and hypotenuse the edge of the pyramid: \[ \text{Slant height} (l) = \sqrt{245^2 - 100^2} \] \[ l = \sqrt{60025 - 10000} \] \[ l = \sqrt{50025} \approx 223.7 \: \text{yd} \]2. **Calculate the Area of One of the Triangle Faces:** - The base of each triangular face = side of square base = 200 yd - Height of each triangular face = slant height (l) ≈ 223.7 yd \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] \[ \text{Area of one triangular face} = \frac{1}{2} \times 200 \times 223.7 \] \[ \text{Area} \approx 22370 \: \text{yd}^2 \]3. **Calculate the Total Surface Area Excluding the Base:** - There are four triangular faces. \[ \text{Total surface area} = 4 \times \text{Area of one triangular face}

12. Imagine a right pyramid with a square base (like an Egyptian pyramid). Suppose that the sides of the square base are all 200 yd long and that the distance from each vertex of the base to the apex of the pyramid (along an edge) is 245 yd. Determine the surface area of the pyramid (not including the base). Explain your reasoning.

12. Imagine a right pyramid with a square base (like an Egyptian pyramid). Suppose that the sides of the square base are all 200 yd long and that the distance from each vertex of the base to the apex of the pyramid (along an edge) is 245 yd. Determine the surface area of the pyramid (not including the base). Explain your reasoning.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter9: Surfaces And Solids

Section9.4: Polyhedrons And Spheres

Problem 23E: The surface of a soccer ball is composed of 12 regular pentagons and 20 regular hexagons. With each...

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