Question

word problem involving the pythagorean theorem in three - dimensions. a tent has the shape of an isosceles triangular prism. the campers add a pole of length b (in feet) for extra support that goes from the top front corner to one of the back bottom corners, as shown in the figure. (the figure is not drawn to scale.) (a) find a. a = ft (b) use your answer to part (a) to find b, the length of the pole. round your answer to the nearest tenth of a foot. b = ft

Explanation:

Step1: Find the base - diagonal length (a)

First, consider the right - angled triangle on the base of the prism with sides 6 ft and 14 ft. By the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where \(c\) is the hypotenuse of the right - angled triangle. Here, \(a = \sqrt{6^{2}+14^{2}}=\sqrt{36 + 196}=\sqrt{232}\).
\[a=\sqrt{4\times58}=2\sqrt{58}\approx 15.23\] ft

Step2: Find the length of the pole (b)

Now, consider a new right - angled triangle with one side as the height of the prism (8 ft) and the other side as the base - diagonal length \(a\approx15.23\) ft. Using the Pythagorean theorem again, \(b=\sqrt{8^{2}+a^{2}}\). Substitute \(a\approx15.23\) ft, then \(b=\sqrt{64 + 232}=\sqrt{296}\).
\[b=\sqrt{4\times74}=2\sqrt{74}\approx17.2\] ft

Answer:

(a) \(a\approx15.2\) ft
(b) \(b\approx17.2\) ft