SOLUTION: From the top of a fire tower, a forest ranger sees his partner on the ground at an angle of depression of 40º. If the tower is 45 feet in height, how far is the partner from the b

Question 1208494: From the top of a fire tower, a forest ranger sees his partner on the ground at an angle of depression of 40º. If the tower is 45 feet in height, how far is the partner from the base of the tower, to the nearest tenth of a foot?
Found 3 solutions by josgarithmetic, ikleyn, math_tutor2020:
Answer by josgarithmetic(39616)   (Show Source): You can put this solution on YOUR website!
You can draw the described triangle, label the parts, and find the needed equation and solve.

45 ft., vertical side
x, horizontal side
40 degree, opposite angle to the vertical side
50 degree , opposite angle of the x side
90 degree, the base, between the 45 ft. and the x ft.
.
.
Law Of Sines:

Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

It looks like @josgarithmetic lost himself in two trees.

Let's wish him to find his right way to get out of these dense thickets.

Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer: 53.6 feet

Explanation
The forest ranger in the tower is at point C.
His eyeline is initially aimed along the dashed line. Then he rotates his view 40 degrees downward since this is the angle of depression.
The other ranger is at point B.

Note in the diagram that 50+40 = 90.
Or you could say 90-40 = 50 so you find that other angle near point C.

Once that 50 degree angle is found, erase the dashed line and erase the 40 degree angle.
We focus entirely on triangle ABC.

It's a right triangle, so you can use the tangent ratio to find the distance from A to B.
tan(angle) = opposite/adjacent
tan(C) = AB/AC
tan(50) = x/45
x = 45*tan(50)
x = 53.628911666739 feet approximately
x = 53.6 feet when rounding to the nearest tenth.
Please make sure that your calculator is set to degrees mode.

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