Solve the problem. Round to the nearest whole unit.
-A 25-inch tall picture hangs on a wall so that its lower edge is 27 inches higher than the eyes of an observer. How far from the wall must the observer stand in order to make the angle subtended by the picture equal to 1/3
Of the angle of elevation of the top?
A) 30 in.
B) 42 in.
C) 36 in.
D) 24 in.

Question

Quizplus · Accepted Answer

The angle subtended by the picture is 1/3 of the angle of elevation of the top. Let x be the distance from the observer to the wall. The top of the picture is 25 + 27 = 52 inches above the observer's eyes. Using tangent for small angle θ, tan(θ) = opposite/adjacent = 52/x, and for the larger angle 3θ, tan(3θ) = opposite/adjacent = 27/x. Using the identity tan(3θ) = 3tan(θ) - tan^3(θ)/(1 - 3tan^2(θ)), and since tan(θ) is small, we can approximate tan(3θ) ≈ 3tan(θ), so 27/x ≈ 3(52/x), which simplifies to x ≈ 36 inches.

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