Find how fast its rising. Need help.

Question

I need help with a calculus problem. You're at a local cemetery and you see a ghost rise up from a grave. The grave is 80 feet away from you with an angle of elevation of the rising ghost is changing at a rate of 3 degrees per second. How fast is the ghost rising when it is 6 feet off the ground. Thank you in advance

Answer 1

Draw a right triangle with right angle at the site of the grave.  

 

Let x = height of ghost after t seconds 

        = length of vertical leg at time t

 

    80 = length of horizontal leg

     

     y = length of hypotenuse at time t

 

     θ = angle of elevation of ghost at time t

 

Given:  dθ/dt = 3°/sec = (3°/sec)(π/180°) = π/60 rad/sec

 

Find:  dx/dt when x = 6 ft

 

Solution:  tanθ = x/80

 

             So, sec²θ(dθ/dt) = (1/80)(dx/dt)

 

             dx/dt = 80sec²θ (dθ/dt)   When x = 6, y² = 6² + 80²

 

                                                                   y = √6436

 

             dx/dt = 80(√(6436))/80)²(π/60)

 

                     ≈ 4.21 ft/sec