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1.) A man 6 ft tall walks at a rate of 5 ft per sec. from a light that is 15 ft above the ground. At what rate is the top of his shadow changing?

2.) An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 mph. Find the rate at which the angle of elevation is changing when the angle is at 60 degrees.

  • Calculus-rates - ,

    Make a diagram showing the lamppost and the man's position any time.
    Join the top of the lamppost to the man's top and extend it to the ground.
    let the distance between the lamppost and the man along the ground be x ft, let the length of his shadow on the ground be y ft.
    Since we have similar triangles, use the ratio
    (x+y)/15 = y/6
    6x + 6y = 15y
    6x = 9y
    2x = 3y
    then 2dx/dt = 3dy/dt
    2(5) = 3dy/dt
    dy/dt = 10/3

    So it appears that the position of the man is irrelevant and the shadow is changing at 10/3 ft/sec
    Had it asked "how fast is his shadow moving ?" you would have added the 5ft/sec to the above answer.

    2nd problem:
    Again, make a diagram showing the plane moving in a line parallel to the ground.
    Let the distance of the plane's path be x miles and the angle of elevation be Ø
    I see that
    tan Ø = 5/x
    x tanØ = 5
    x sec^2 Ø dØ/dt + tanØ dx/dt = 0

    so when Ø = 60° , x = 5/√3

    (5/√3)(1/3) dØ/dt + √3(600) = 0
    dØ/dt = -1080 radians/hour
    or 3/10 radians/sec , (I divided by 3600 to get to seconds)

    (since derivatives in trig only work with radians, the answer obtained would be in radians)

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