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 the length of the diagonal of the base square

The base of the square - pyramid is a square with side - length 5 m. The length of the diagonal of a square with side - length \(s\) is given by \(d = \sqrt{2}s\). Here, \(s = 5\) m, so the length of the diagonal of the base square \(d=\sqrt{2}\times5 = 5\sqrt{2}\text{ m}\). The distance from the center of the square base to a corner of the base \(a=\frac{d}{2}\). Since \(d = 5\sqrt{2}\text{ m}\), then \(a=\frac{5\sqrt{2}}{2}\text{ m}\approx\frac{5\times1.414}{2}=3.535\text{ m}\approx3.5\text{ m}\).

Step2: Use the Pythagorean theorem to find \(b\)

We know the height of the pyramid is 12 m and the distance from the center of the base to a corner of the base is \(a\approx3.5\text{ m}\). By the Pythagorean theorem, for a right - triangle with legs \(a\) and height \(h = 12\) m and hypotenuse \(b\), \(b=\sqrt{a^{2}+h^{2}}\). Substituting \(a\approx3.5\text{ m}\) and \(h = 12\) m, we get \(b=\sqrt{(3.5)^{2}+12^{2}}=\sqrt{12.25 + 144}=\sqrt{156.25}=12.5\text{ m}\).

Answer:

a) \(3.5\)
b) \(12.5\)