Question #172070

Sighting from a helicopter: A helicopter hovers at an altitude that is 1000 feet above a mountain peak of altitude 5210 feet. A second, taller peak is viewed from both the mountaintop and the helicopter. From the helicopter, the angle of depression is 43 degrees,and from the mountaintop, the angle of elevation is 18 degree . How far apart are the mountain peaks? What is the altitude of the taller peak?


1
Expert's answer
2021-03-17T12:24:41-0400

Let A,B be mountain peaks\text{Let A,B be mountain peaks}

C be helicopter’s location\text{C be helicopter's location}

let B be such a point that:\text{let B be such a point that:}

ADis the distance between the mountain peaksAD \, \text{is the distance between the mountain peaks}

BDis the difference in height between mountain peaksBD \,\text{is the difference in height between mountain peaks}

BDA=90°\angle{BDA}=90\degree

CAD=90°\angle{CAD}=90\degree

from the conditions of the problem\text{from the conditions of the problem}

BAD=18°\angle{BAD}=18\degree

ACB=43°\angle{ACB}=43\degree

CA=1000CA = 1000

find angles\text{find angles}

α=CAB=CADBAD=90°18°=72°\alpha=\angle{CAB}= \angle{CAD}-\angle{BAD}=90\degree-18\degree=72\degree

γ=ACB=43\gamma=\angle{ACB}= 43^{\circ}

β=CBA=180°CABBCA=\beta=\angle{CBA}=180\degree-\angle{CAB}-\angle{BCA}=

180°(72°+43°)=65°180\degree-(72\degree+43\degree)=65\degree

by the sine theorem ABC\text {by the sine theorem }\triangle{ABC}

ABsinγ=ACsinβ\frac{AB}{\sin{\gamma}}=\frac{AC}{\sin{\beta}}

AB=ACsinβsinγAB=\frac{AC}{\sin{\beta}}\sin{\gamma}

from ABD\text{from } \triangle{ABD}

BD=ABsinBAD=ACsinβsinγsinBADBD=AB*\sin{\angle{BAD}}=\frac{AC}{\sin{\beta}}\sin{\gamma}*\sin\angle{BAD}

BD=1000sin65°sin43°sin18°BD = \frac{1000}{\sin65\degree}\sin43\degree\sin18\degree\approx 232.5

AD=ABcosBAD=ACsinβsinγcosBADAD=AB*\cos{\angle{BAD}}=\frac{AC}{\sin{\beta}}\sin{\gamma}*\cos\angle{BAD}

AD=1000sin65°sin43°cos18°715.7AD = \frac{1000}{\sin65\degree}\sin43\degree\cos18\degree\approx715.7

then the height of the highest mountain peak\text {then the height of the highest mountain peak}

h=BD+5210=232.5+5210=5442.5 fth = BD+5210= 232.5+5210= 5442.5\ ft


Answer: 715.7 ft distance between mountain peaks

5442.5 ft the height of the highest mountain peak















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