Question

an architecture firm is designing a pavillion 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 = square ) 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 = square ) m
(figure: a square pyramid with base side 18 m (split into two 9 m segments), a height of 12 m, a right triangle with legs 9 m and a (blue), and another right triangle with legs a and 12 m (red) for edge b.)

Explanation:

Step1: Find \( a \) using Pythagorean theorem

The triangle for \( a \) is a right triangle with legs 12 m and 9 m. By Pythagorean theorem, \( a = \sqrt{12^2 + 9^2} \).
Calculate \( 12^2 = 144 \), \( 9^2 = 81 \), sum is \( 144 + 81 = 225 \). So \( a = \sqrt{225} = 15 \) m.

Step2: Find \( b \) using Pythagorean theorem

Now, for \( b \), the right triangle has legs \( a = 15 \) m and 9 m. So \( b = \sqrt{15^2 + 9^2} \).
Calculate \( 15^2 = 225 \), \( 9^2 = 81 \), sum is \( 225 + 81 = 306 \). Then \( b = \sqrt{306} \approx 17.5 \) m (rounded to nearest tenth).

Answer:

(a) \( a = \boldsymbol{15} \) m
(b) \( b \approx \boldsymbol{17.5} \) m