At what rate is the shadow​ decreasing?

Question

On a morning of a day when the sun will pass directly overhead, the shadow of a 60-ft building on level ground is 25 ft long. At the moment in question, the angle theta the sun makes with the ground is increasing at the rate of 0.26degrees/min. At what rate is the shadow decreasing? Remember to use radians in your calculations. Express your answer in inches per minute.

Answer 1

x = length of shadow

h = height of building = 60 ft

 

Draw and label a diagram, and see that

 

tanθ = h/x 

 

x = h/tanθ = h cosθ / sinθ 

 

dx/dt = (-h/sin²θ) dθ/dt

 

When x = 25,

 

θ = tan^-1(60/25) = 67.38°

h = 60 ft = constant

dθ/dt = 0.26 degrees/min

 

Now convert h to inches, convert degrees/min to radians/min, plug the converted quantities into 

 

dx/dt = (-h/sin²θ) dθ/dt,

 

and get your answer in inches per minute.  Note dx/dt < 0, indicating decreasing x.  You should be able to finish from here.