Question

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=squaremathrm{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 = squaremathrm{ft})

Explanation:

Step1: Find the length of \(a\) using the Pythagorean theorem in the base - rectangle

The base of the tent has a rectangle with sides 6 ft and 14 ft. To find the diagonal of the base rectangle (which is \(a\)), we use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 6\) and \(b=14\).
\[a=\sqrt{6^{2}+14^{2}}=\sqrt{36 + 196}=\sqrt{232}\]
\[a=\sqrt{4\times58}=2\sqrt{58}\approx 15.23\] ft

Step2: Find the length of \(b\) using the Pythagorean theorem considering the height of the tent

We know the height of the tent is 8 ft and we just found \(a\approx15.23\) ft. Now, to find \(b\) (the length of the pole), we use the Pythagorean theorem again. Let the height of the tent be \(h = 8\) ft and the base - diagonal be \(a\). Then \(b=\sqrt{a^{2}+h^{2}}\).
\[b=\sqrt{(2\sqrt{58})^{2}+8^{2}}=\sqrt{232 + 64}=\sqrt{296}\]
\[b=\sqrt{4\times74}=2\sqrt{74}\approx17.2\] ft

Answer:

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