Calculus rate of change problem?
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Could someone solve this problem with detailed work?
Thanks!
An observer stands 150 meters from a fireworks display rocket which is fired directly upward. When the rocket reaches a height of 200 meters, it is traveling at a speed of 12 meters/second. At what rate is the angle of elevation formed with the observer increasing at that instant?
Calculus rate of change problem?Mmmk so let's get together our terms and what we know here...
y = height of rocket
x = horizontal distance of observer from launchpoint
dy/dt = speed of rocket
Now, we want to know the change in the angle of elevation, theta, that is formed with respect to the observer. Now, we notice that if at any time, t, we draw a line from the observer to the launchpoint, from the rocket to the launchpoint, and from the rocket to the observer, we get a right triangle. We then notice that the angle of elevation can be defined in terms of the rocket's path, where
tan(theta) = y/x
We note that x is constant, so we'll simplify to
tan(theta) = y/150
We also note that theta at the moment we are interested in is arctan(200/150) = .927 rad
Now, we differentiate each side of the equation with respect to t
tan(theta) = y/150
[d(theta)/dt]*sec^2(theta) = (1/150)*dy/dt
solve for d(theta)/dt
d(theta)/dt = cos^2(theta)/150 * dy/dt
now we simply plug in our variables
d(theta)/dt = [cos^2(.927)/150] * 12 = .0288 rad/s
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