Related Rates Problem :(

Question

A balloon is rising at a rate of 5 meters per second from a point on the ground 46 meters from an observer. Find the rate of change of the angle of elevation from the observer to the balloon when the balloon is 32 meters above the ground.

Answer 1

A balloon is rising at a rate of 5 meters per second

=> rising = positive (increasing height)

=> dh/dt = 5 m/sec

Ballon

h (meter)

Theta

Observer--------------------------------------------------

46m

tan(theta) = h/46

if we take a derivative,

sec^2(theta) * d(theta)/dt=1/46 * dh/dt,

[by Pythagorean thm: hypotenuse ^2 = one leg^2 + another leg^2

hyp^2=46^2+32^2 => h= 56.0357029]

sec^2 (theta) d(theta)/dt = 1/46 * 5

d theta/dt = 5/46 * cos^2 (theta)= 0.07324840764 rad/sec (costheta=46/56.035]