If a child is climbing on Great Forest Mountain and she is in the middle, her angle of elevation to a second Mountain is 42 and angle of depression to that mountain is 32. The height of the second...

If a child is climbing on Great Forest Mountain and she is in the middle, her angle of elevation to a second Mountain is 42 and angle of depression to that mountain is 32. The height of the second Mountain is 2000ft. What is the length of what the child is looking at the second mountain?

embizze | Certified Educator

We are asked to find the distance between two mountains (base to base) given that the angle of elevation for an observer to the top is 42 degrees and the angle of depression to the base is 32 degrees and the height of the target mountain is 2000 feet. (Please see the diagram attached.)

Setting the height above the horizontal sight line as x feetthe height of the mountain below the horizontal sight line is 2000-x feet. We now have two right triangles. Let d be the distance between the mountains.

The triangle above the horizontal sight line has an acute angle of 42 degrees, the side opposite is labelled x, and the side adjacent is labelled d.

Using the tangent ratio we have tan(42)=x/d.

The triangle below the horizontal sight line has an acute angle of 32 degrees with the side opposite labelled 2000-x, and the side adjacent labelled d.

Using the tangent ratio we get tan(32)=(2000-x)/d.

Solving each equation for we get d=x/(tan(42))=(2000-x)/(tan(32)).

Then xtan(32)=(2000-x)tan(42)

xtan(32)+xtan(42)=2000tan(42)

x(tan(32)+tan(42))=2000tan(42)

x=(2000tan(42))/(tan(32)+tan(42))

So x is approximately 1180.65. Substituting we find d is approximately 1311.24

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The mountains are approximately 1311 feet apart

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