Question

A balloon rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer. Find the rate of change of the angle of elevation when the balloon is 25 feet above the ground.

I have d(theta)/dt=(1/60)cos^2(theta)(8)
How do I find theta?

Answer 1

tan T = h/60

d tan T /dt = dT/dt sec^2 T = (1/60)dh/dt

dT/dt = (1/60) cos^2 T dh/dt

tan T = 25/60
T = 22.6 deg
so cos^2 T = .852

dT/dt = (1/60)(.852) (8)

in radians !

Answer 2

Let the height of the ballon be h ft
then tanØ = h/60
h = 60tanØ

dh/dt = 60 sec^2 Ø dØ/dt

we know dh/dt = 8
when h = 25
tanØ = 25/60 = 5/12
construct a right-angled triangles with opposite side 5 and adjacent side 12.
By Pythagoras the hypotenuse is 13
so without even finding the actual angle we see that
cosØ = 12/13
cos^2 Ø = 144/169

Picking up where you left off
dØ/dt = (8/60)(144/169)
= <b>96/845 rad/s</b> or
= appr .1136 rad/s

you were so close