The height in feet of a rocket is fired straight up from the surface of the wate
ID: 2841899 • Letter: T
Question
The height in feet of a rocket is fired straight up from the surface of the water of the Barents Sea by an inattentive Russian Typhoon class missile submarine in t seconds is given by s(t) = -16t^2 +3200t Exactly 3 miles away a quietly stalking American Virginia class submarine is watching the launch through its periscope eye which is exactly at the same level as the water. A.) Find the rate of change of the angle of elevation ? of the line of sight from the periscope and the rocket at t =1 second. B.) Find the maximum value the angle of elevation ? attains. Please show work so that I can understand and full points will be rewarded. Thank you :)Explanation / Answer
Since this equation is giving us height depending on time, if we differentiate with repsect to time, we get a formula for velocity. So, s'(t)=-32t+3200. Then, the instantaneous velocity at t=1 is 3200-32*1=3168 feet per second. Now, as we are trying to solve for rate of change, we set up a right triangle where the vertical axis (s(t)) is height, and the horizontal axis (x) is the distance away from the start of the objects movement. Note that tan (theta) = s(t)/x. As x is always 3 miles, or 15840 feet, we have tan (theta)=s(t)/15840. If we solve for theta at at time of 1, we get theta = 11.37 degrees Keep this in your head for now.
Now, we differentiate both sides with respect to t.
(d theta/dt)sec^2 (theta)=1/15840 (d s(t)/dt)
We solve for (d theta/dt), as this is the rate of change of theta:
(d theta/dt)=1/sec^2 (theta) * 1/15840 *(d s(t)/dt)
Since we have theta=11.37..., and (d s(t)/dt)=3168, we have
(d theta/dt)= approximately 0.1922.
To find the maximum value of theta, we must first find the maximum height of the rocket. This happens when the velocity of the rocket becomes zero.
Therefore, we have -32t+3200=0, and t=100.
Therefore, at t=100, we have the maximal height.
We plug in 100 into s(t) to get s(100)=-16*100^2+3200*100=160000.
Then, we have tan(theta)=160000/15840, so theta=arctan(160000/15840)=84.35 degrees.
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