Question

The angle of depression to the near side of a lake, measured from a balloon 2500 feet above the lake is 43°. The angle of depression to the far side of the lake is 27°. Find the width of the lake, round to the nearest tenth Width of the lake: feet​

Answer 1

7587.4 ft

Explanation:

You want the width of a lake if the angle of depression to the near shore from a balloon 2500 ft above the lake is 43°, and to the far shore is 27°.

Geometry

The first attachment shows the geometry of the problem when the balloon is above the lake. In this geometry, the tangents of the angles of depression are ...

Tan = Opposite/Adjacent

tan(43°) = OB/ON

tan(27°) = OB/OF

Solution

The width of the lake is ON +OF, so is ...

ON = OB/tan(43°)

OF = OB/tan(27°)

NF = OB·(1/tan(27°) +1/tan(43°)) = 2500·(1/tan(27°) +1/tan(43°))

NF ≈ 7587.4 . . . . feet

The width of the lake is about 7587.4 feet.

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Additional comment

This problem identifies the position of the balloon as being "above the lake." Above, we took that literally. Often these problems have the observation point off to one side of the distance being observed, meaning the balloon would be above the near shore of the lake. If that is the intended geometry, the width of the lake is computed from the difference of the cotangents, rather than their sum. That distance is 2225.6 feet.

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