Question

justin is flying two kites. he has 101 feet of string out to one kite and 114 feet out to the other kite. the angle formed by the two strings is 32° 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.

Explanation:

Step1: Identify the Law to Use

We have two sides of a triangle (101 ft, 114 ft) and the included angle (32°), so we use the Law of Cosines. The formula for the Law of Cosines is \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( a = 101 \), \( b = 114 \), \( C = 32^\circ \), and \( c \) is the distance between the kites.

Step2: Substitute Values into the Formula

First, calculate \( a^2 \), \( b^2 \), and \( 2ab\cos(C) \).
\( a^2 = 101^2 = 10201 \)
\( b^2 = 114^2 = 12996 \)
\( 2ab = 2\times101\times114 = 23028 \)
\( \cos(32^\circ) \approx 0.848048 \) (calculated to four decimal places)
\( 2ab\cos(C) = 23028\times0.848048 \approx 19528.5343 \)
Now, substitute into the Law of Cosines:
\( c^2 = 10201 + 12996 - 19528.5343 \)
\( c^2 = 23197 - 19528.5343 = 3668.4657 \)

Step3: Solve for \( c \)

Take the square root of \( c^2 \):
\( c = \sqrt{3668.4657} \approx 60.567 \) (calculated to four decimal places)
Round to the nearest tenth: \( c \approx 60.6 \)

Answer:

60.6