Question

an architecture firm is designing a pavilion in the shape of a square pyramid. they plan to add a string of lights of length b (in meters) to each of the edges that meet at the top. in the figure, the edge with length b shows the placement of one string of lights. (the figure is not drawn to scale.) (a) find a. a = m (b) use your answer to part (a) to find b, the length of one string of lights. round your answer to the nearest tenth of a meter. b = m

Explanation:

Step1: Find \(a\) using the Pythagorean theorem in the base - related right - triangle

The base of the square pyramid has a right - triangle with legs of length 3m and 4m. By the Pythagorean theorem \(c^{2}=x^{2}+y^{2}\), where \(x = 3\) and \(y = 4\). So \(a=\sqrt{3^{2}+4^{2}}\).
\[a=\sqrt{9 + 16}=\sqrt{25}=5\]

Step2: Find \(b\) using the Pythagorean theorem in the larger right - triangle

We now have a right - triangle with one leg \(a = 5\)m and the other leg 3m. Using the Pythagorean theorem \(b=\sqrt{a^{2}+3^{2}}\). Substitute \(a = 5\) into the formula: \(b=\sqrt{5^{2}+3^{2}}=\sqrt{25 + 9}=\sqrt{34}\approx5.8\)

Answer:

(a) \(a = 5\) m
(b) \(b\approx5.8\) m