Respuesta :
The height of the tree, which Pat needs to determine, before cutting it down to be sure that it will not fall on a nearby fence is 111 ft.
What is right angle triangle property?
In a right angle triangle, the ratio of the opposite side to the base side is equal to the tangent angle between them.
[tex]\tan\theta=\dfrac{b}{a}[/tex]
Here, (b) is the opposite side, (a) is the base side.
Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence.
The angle of elevation of the tree from one position on a flat path from the tree is
[tex]H=40^o[/tex]
From a second position,
[tex]I=60\rm\; ft[/tex]
Farther along this path it is,
[tex]b=30^o[/tex]
Let the distance between tree and first position is x. Thus, from the trigonometry,
[tex]\tan 40=\dfrac{h}{x}\\h=x\tan 40[/tex] ....1
The distance between tree and second position is (x+60) ft. Thus, again from the trigonometry,
[tex]\tan 30=\dfrac{h}{x+60}\\h=(x+60)\tan 40[/tex] ......2
Compare the equation 1 and 2,
[tex]x\tan (40)=(x+60)\tan 30\\0.839x=0.577x+34.64\\0.839x-0.577x=34.64\\x=\dfrac{34.64}{0.262}\\x=132.25\rm\; ft[/tex]
Put this value in equation 1 as,
[tex]h=(132.25)\times \tan40\\h\approx111\rm\;ft[/tex]
Thus, the height of the tree, which Pat needs to determine, before cutting it down to be sure that it will not fall on a nearby fence is 111 ft.
Learn more about the right angle triangle property here;
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