Find the rate of change of the angle of elevation dθdx when x=272 feet

Question

A building that is 225 feet tall casts a shadow of various lengths  as the day goes by. An angle of elevation (Theta) is formed by lines from the top and bottom of the building to the tip of the shadow, Find the rate of change of the angle of elevation dθ/dx when x=272 feet

any help would be greatly appreciated :)

Answer 1

Find the rate of change of the angle of elevation dθdx when x=272 feet

A building that is 225 feet tall casts a shadow of various lengths  as the day goes by. An angle of elevation (Theta) is formed by lines from the top and bottom of the building to the tip of the shadow, Find the rate of change of the angle of elevation dθ/dx when x=272 feet

any help would be greatly appreciated :)

Hello Daniel,

Use tanθ = opp / adj

or

tanθ = buidling height / shadow length

tanθ = 225 / x

find derviative:

d/dx [ tanθ = 225 / x ]

sec²θ*dθ/dx = -225 / x²

Solve for dθ/dx

dθ/dx = -225 / (x²*sec²θ)

Sub x = 272, θ = tan^-1(225/272) = 0.691rad

dθ/dx = -225 / (272²*sec²(0.691))

dθ/dx = -0.00180598 rad/foot