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An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 miles per hour. Find the rate at which the angle of elevation is changing when the angle is pi/6.
Heya Bobby,
If you consider a right angle triangle as illustrated above, with the observer at O and the plane at P, you can see the relationship between the angle of elevation (\(\displaystyle \theta\)) and the distance to the plane (x) is given by:
\(\displaystyle tan \theta = {5 \over x}\)
By implicit differentiation with respect to time:
\(\displaystyle sec^2 \theta {d \theta \over dt} = -{5 \over x^2} {dx \over dt}\)
\(\displaystyle {d \theta \over dt} = -{5 \over x^2} {dx \over dt} {1 \over sec^2 \theta }\)
Remember \(\displaystyle sec \theta = {1 \over cos \theta}\)
Therefore, \(\displaystyle {d \theta \over dt} = -{5 \over x^2} {dx \over dt} cos^2 \theta\)
\(\displaystyle {dx \over dt} = -600 mi.hr^{-1}\)
\(\displaystyle \theta = {\pi \over 6}\)
And you can determine what x must be for \(\displaystyle \theta\) to equal \(\displaystyle {\pi \over 6}\)
Substitute these values in and you will get your answer.
If you consider a right angle triangle as illustrated above, with the observer at O and the plane at P, you can see the relationship between the angle of elevation (\(\displaystyle \theta\)) and the distance to the plane (x) is given by:
\(\displaystyle tan \theta = {5 \over x}\)
By implicit differentiation with respect to time:
\(\displaystyle sec^2 \theta {d \theta \over dt} = -{5 \over x^2} {dx \over dt}\)
\(\displaystyle {d \theta \over dt} = -{5 \over x^2} {dx \over dt} {1 \over sec^2 \theta }\)
Remember \(\displaystyle sec \theta = {1 \over cos \theta}\)
Therefore, \(\displaystyle {d \theta \over dt} = -{5 \over x^2} {dx \over dt} cos^2 \theta\)
\(\displaystyle {dx \over dt} = -600 mi.hr^{-1}\)
\(\displaystyle \theta = {\pi \over 6}\)
And you can determine what x must be for \(\displaystyle \theta\) to equal \(\displaystyle {\pi \over 6}\)
Substitute these values in and you will get your answer.