Calculus 1, Related Rates, Circle Problem?

Calculus 1. Related Rates. Chetna is racing around a circular track, starting at the easternmost edge and running counterclockwise at a constant pace of 40 seconds per lap. At the center of the track is a lamp, and 60 meters east of the lamp is a wall that runs north-south. When Chetna is one-twelfth of the way around the track, how fast is her shadow moving along the wall?

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  • 1 month ago

    Full track is 2piRadius

    2pi*60 = 377m

    Circunference over 40 per lap is speed.

    377/40 = 9.24 meters per second

  • She runs 2pi radians in 40 seconds

    2pi/40 = pi/20

    dT/dt = pi/20 radians/second

    The wall is 60 meters away from the center of the circle and T = pi/6 (because 2pi/12 = pi/6)

    If we say that the point on the wall closest to the lamp is O, then we can say that the shadow touches the wall at some point y, where y is y meters north or south of O.

    So you have a right triangle.  x = 60.  y = ?.  y/x = tan(T)

    y = x * tan(T)

    y = 60 * tan(T)

    Derive with respect to time

    dy/dt = 60 * sec(T)^2 * dT/dt

    T = pi/6 , dT/dt = pi/20

    dy/dt = 60 * sec(pi/6)^2 * (pi/20)

    dy/dt = 3 * pi / cos(pi/6)^2

    dy/dt = 3 * pi / (sqrt(3)/2)^2

    dy/dt = 3 * pi / (3/4)

    dy/dt = 3 * pi * 4 / 3

    dy/dt = 4 * pi

    The shadow is moving along the wall at 4pi meters per second at that moment.

  • 1 month ago

    Let x = how many degrees Chetna has run around the track (2*pi radians per lap)

    Let y = distance from the shadow to the point on the wall that's closest to the circletan(x) = y/60

    y = 60*tan(x)

    dy/dx = 60*sec^2(x)

    dx/dt = 1/40 laps per second, that's 2*pi/40 = pi/20 radians per second

    dy/dt = dy/dx * dx/dt

    dy/dt = 60*sec^2(x) * (pi/20)

    dy/dt = 3*pi*sec^2(x)

    When Chetna is 1/12 of the way around the track, x = 2*pi/12 = pi/6 radians, so we have:

    dy/dt = 3*pi*sec^2(pi/6)

    dy/dt = 3*pi*(2/sqrt(3)) ^2

    dy/dt = 3*pi*4/3

    dy/dt = 4*pi

    dy/dt =~ 12.566370614359172953850573533118 meters per second

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