Question

two tugboats that are a = 135 ft apart pull a barge, as shown. if the length of one cable is b = 204 ft and the length of the other is c = 225 ft, find the angle formed by the two cables (round your answer to the nearest degree.)

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\). Here, \(a = 135\), \(b = 204\), \(c=225\). We need to solve for \(A\).
First, substitute the values into the formula:
\(135^{2}=204^{2}+225^{2}-2\times204\times225\times\cos A\)
\(18225=41616 + 50625-91800\cos A\)
\(18225=92241-91800\cos A\)

Step2: Rearrange the equation to solve for \(\cos A\)

\(91800\cos A=92241 - 18225\)
\(91800\cos A=74016\)
\(\cos A=\frac{74016}{91800}\)
\(\cos A = 0.8062745\)

Step3: Find the angle \(A\)

\(A=\cos^{-1}(0.8062745)\)
Using a calculator, \(A\approx36^{\circ}\)

Answer:

\(36^{\circ}\)