Question

susan is flying two kites. she has 101 feet of string out to one kite and 117 feet out to the other kite. the angle formed by the two strings is 31° as shown in the figure below. find the distance between the kites. carry your intermediate computations to at least four decimal places. round your answer to the nearest tenth of a foot. feet

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \( c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 101\), \(b = 117\), and \(C=31^{\circ}\).
So, \(c^{2}=101^{2}+117^{2}-2\times101\times117\times\cos(31^{\circ})\).
First, calculate \(101^{2}=10201\), \(117^{2}=13689\).
Then, \(\cos(31^{\circ})\approx0.8572\).
And \(2\times101\times117 = 2\times11817=23634\).

Step2: Compute \(c^{2}\)

\(c^{2}=10201 + 13689-23634\times0.8572\).
\(10201+13689 = 23890\).
\(23634\times0.8572\approx23634\times0.8572 = 20269.0648\).
\(c^{2}=23890-20269.0648=3620.9352\).

Step3: Find \(c\)

\(c=\sqrt{3620.9352}\approx60.2\).

Answer:

\(60.2\) feet