Calculus Problems On Related Rates

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Related Rates Calculus Problem?

Bugs and Daffy finished their final act on the Bugs and Daffy Show by dancing off the stage with a spotlight covering their every move. If they are moving along a straight path at a speed of 4 ft/s, and the spotlight is 20 ft away from this path, what rate is the spotlight rotating when they are 15 feet from the point on the path closest to the spotlight?

we have a right triangle…

let x = distance from the actors to the point closest to the spotlight on the path
let θ = angle opposite x

then

tanθ = x/20

differentiating…
sec² θ dθ/dt = (1/20)dx/dt

now.. if x = 15
tan θ = 15/20 = 3/4
then sec θ = 5/4

then
25/16 dθ/dt = (1/20)(4)

dθ/dt = 16/125 radians per second.

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Calculus: Related Rates 1

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