College Calculus 1

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A runner sprints around a circular track of radius 150 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 300 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 300 m? (Round to 2 decimal places.

My teacher tried helping me with this problem and only confused me more. I drew a diagram. I used the formula c^2=a^2+b^2-2abcosC. Right now I have 2L(dL/dt)=2(150)(300)sin(angle)(dangle/dt) I think (dangle/dt) is 7/150. This would give you (solving for dL/dt:
(dL/dt)= (45000sin(angle)(7/100))/300
HELP!!!!!

  • College Calculus 1 -

    I suggest the following approach.

    Let the friend (outside of the tracks) be at the origin.

    Express the x and y coordinates of the runner in terms of t, for example,
    X(t)=rcos(st/r)+d
    Y(t)=rsin(st/r)

    s=speed in m/s
    r=radius
    d=given distance

    Express the distance between the two persons in terms of t:
    L(t)=√(X(t)²+Y(t)²)
    Differentiate L(t) with respect to t, which gives L'(t), the rate of change of distance as a function of t.

    Finally, find the times at which the two persons are 300m apart, namely, solve for t at which
    L(t)=300.

    The answers are t1=39 and t2=96 seconds approx.
    L'(t1) and L'(t2) are the required answers.

    I get about ±6.8 m/s, which is reasonable considering that the runner runs at 7m/s and the points in question are not far from the points of tangency (where L'(t) should give ±7m/s).

  • College Calculus 1 -

    I have no fricken clue. That's why I fricken hatee MATH. LIKE HOLY FRICK! WHY THE FRICK DO WE HAVE TO LEARN?

  • College Calculus 1 -

    JEEZZZ-,-
    dont understand-.-"
    it will be easier to learn by talking...
    im confused with the signs hahaha!!!

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