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Math Help - Related rates: urgent

  1. #1
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    Exclamation Related rates: urgent

    Can anyone help
    A baseball diamond has the shape of a square with sides 90 feet long. A player 30 feet from third base is running at a speed of 28 feet per second. At what rate is the player’s distance s from the home plate?
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  2. #2
    MHF Contributor
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    Quote Originally Posted by ml_2007 View Post
    Can anyone help
    A baseball diamond has the shape of a square with sides 90 feet long. A player 30 feet from third base is running at a speed of 28 feet per second. At what rate is the player’s distance s from the home plate?
    The question is not very clear, but I assume that " A player 30 feet from third base is running ..." means the player is running from 2nd base to 3rd base.

    With that assumption, imagine, or draw the figure on paper. It is a right triangle, with these:
    ---one leg = x ---------the portion of 2nd-to-3rd-base side, from the 3rd base.
    ---other leg = 90 ft ----the 3rd-to-home-base side.
    ---hypotenuse = s -----distance from runner to the home base.

    By Pythagorean theorem,
    s^2 = x^2 +90^2
    Differentiate both sides with respect to time t,
    2s(ds/dt) = 2x(dx/dt)
    s(ds/dt) = x(dx/dt) ------------(1)

    Now, in (1) we know that:
    dx/dt = -28 ft/sec ----negative because x is getting shorter with time t.
    x = 30 ft
    We are looking for ds/dt.
    But we do not know the s. So we find the s at that instant.

    When x=30 ft,
    s^2 = 30^2 +90^2
    s^2 = 9000
    s = 30sqrt(10)

    So, plug those into (1),
    [30sqr(10)](ds/dt) = 30(-28)
    ds/dt = -28/sqrt(10)
    ds/dt = -8.854 ft/sec ----negative, so s is getting shorter with time also.

    Therefore, the distance of the player from the home base at that instant is decreasing at the rate of 8.854 ft/sec. ---------answer.
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