Answer:
21.53 ft/s
Explanation:
You want the rate of movement along a wall of a rotating light projected from 17 ft away when the rotation rate is 1 revolution per 5 seconds and the angle is 5° from perpendicular.
Solution
The distance x from the point closest to the light is ...
x = (17 ft)·tan(θ)
where θ is (2π)(5°/360° +t/5) . . . . . . defining t=0 at the point of interest
The rate of change is ...
x' = (17 ft/s)·sec(θ)²(2π/5)
At t = 0, θ = π/36, so this is ...
x'(0) = 6.8π/cos(π/36)² ft/s ≈ 21.53 ft/s
The light is moving at about 21.53 ft/s when the angle is 5°.
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Additional comment
Using a table of derivatives or other means, you can find d(tan(u)) = sec(u)²·du. For the purpose of figuring this with a scientific calculator, sec(x) = 1/cos(x).
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