5. a large monument with height of 2351 m is located exactly halfway between points k and l. michael is…
5. a large monument with height of 2351 m is located exactly halfway between points k and l. michael is standing at point m and finds that he is 14.2 km from l and 17.1 km from k. he also measures the angle between k and l to be 47°. determine the angle of elevation to the top of the monument, measured from point m.
Answer
Explanation:
Step1: Use the Law of Cosines
The Law of Cosines formula is (c^{2}=a^{2}+b^{2}-2ab\cos C). Let (a = 14.2), (b=17.1), and (C = 47^{\circ}). First, find the distance (d) from (M) to the base of the monument (since the monument is halfway between (K) and (L)). [ \begin{align*} d^{2}&=14.2^{2}+17.1^{2}-2\times14.2\times17.1\times\cos47^{\circ}\ d^{2}&=201.64 + 292.41-485.64\times0.682\ d^{2}&=494.05-331.21\ d^{2}&=162.84\ d&=\sqrt{162.84}\approx 12.76\space km = 12760\space m \end{align*} ]
Step2: Use the tangent function
The tangent of the angle of elevation (x) is given by (\tan x=\frac{\text{opposite}}{\text{adjacent}}). Here, the opposite side is the height of the monument ((h = 2351\space m)) and the adjacent side is (d\approx12760\space m). [ \tan x=\frac{2351}{12760}\approx0.1842 ] [ x=\arctan(0.1842)\approx 10.4^{\circ} ]
Answer:
The angle of elevation is approximately (10.4^{\circ})