Question

larry is reroofing his house. each side of the roof measures 14 feet from the eave to the ridge. the horizontal distance between the eaves is 20 feet. the measure of the angle between the two sides of the roof is approximately

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines formula is \( c^{2}=a^{2}+b^{2}-2ab\cos C\). Here, \(a = 14\), \(b = 14\), \(c = 20\), and \(C=x\).
Substitute the values into the formula: \(20^{2}=14^{2}+14^{2}-2\times14\times14\times\cos x\).

Step2: Simplify the equation

First, calculate the squares: \(400 = 196+196 - 392\cos x\).
Then, add \(196 + 196\): \(400=392 - 392\cos x\).
Rearrange the equation: \(392\cos x=392 - 400\).
So, \(392\cos x=-8\).

Step3: Solve for \(\cos x\)

\(\cos x=\frac{- 8}{392}=-\frac{2}{98}=-\frac{1}{49}\approx - 0.0204\).

Step4: Find \(x\)

\(x=\cos^{-1}(-0.0204)\). Using a calculator, \(x\approx91.2^{\circ}\).

Answer:

\(91.2\)