A calculus problem...?
I figured this problem out using sin law but I'm not sure if it's right...can someone do it out and tell me if they get 7 degrees per minute?
"A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off point. At the moment the range finder's elevation angle is pi/4, the height of the balloon is increasing at the rate of 140ft/min. How fast is the elevation angle increasing at that moment?"
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My math skills are a little rusty, but I really want to give this a shot: Feel free to disagree
R: range finder
H: hot air balloon.
theta: angle formed
t: time
We want: theta_dot: the derivative of theta with respect to time: How fast the elevation angle increasing when theta = pi/4
...................................H
....................................I
....................................I Assume that H gain altitude at a constant rate.
....................................I a = 140 t;
....................................I
........Theta................... I
R---------------------------------Origin
.............500 ft
Tan (theta) = a / 500 = 140*t/500
Derive left and right hand side with respect to time:
theta_dot * 1/(cos(theta))^2 = 140 / 500
theta_dot = 140 / 500 *cos(theta)^2
when theta = pi/4
theta_dot = 140 / 500 *cos(pi/4)^2 = 140 / 1000 = 0.14 rad per min
theta_dot = 16.04 degree per minute