Results 1 to 3 of 3

Math Help - related rates

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    68

    related rates

    I had a hard time figuring out this problem. I tried different formulas and plugged in the numbers but I didn't get the correct answer. Please help me. Thanks a lot.
    1. A baseball diamond is a square with sides 90 ft. At the moment a batter hits the ball, a runner from first base starts running to second base with a speed of 22 ft/sec and a runner from second base starts running to third base with a speed of 28 ft/sec. At what rate is the distance between these two runners changing 1 second later . Is this distance increasing or decreasing?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2008
    Posts
    76
    I am new at this stuff too, but if you google "related rates baseball diamond" there are several examples. I think the baseball diamond is used commonly in these type questions.

    Hope you find some help.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,547
    Thanks
    539
    Hello, jsu03!

    I hope you had a good diagram.


    1. A baseball diamond is a square with sides 90 ft. At the moment a batter hits the ball,
    a runner from first base starts running to second base with a speed of 22 ft/sec
    and a runner from second base starts running to third base with a speed of 28 ft/sec.
    At what rate is the distance between these two runners changing 1 second later?
    Is this distance increasing or decreasing?
    Code:
                        C
                     y  *
                   D  *   * 90-x
                    o       *  B
                  *           o
                *               * x
              *                   *
            *                       * A
              *                   *
                *               *
                  *           *
                    *       *
                      *   *
                        *

    At time t, the runner on 1st has run x feet from A\text{ to }B,
    . . where \frac{dx}{dt} = 22 ft/sec. .Note that: BC \:=\:90-x

    At the same time, the runner on 2nd has run y feet from C\text{ to }D,
    . . where \frac{dy}{dt} = 28 ft/sec.

    Draw line segment BD, and let z = BD.


    Since BCD is a right triangle: . z^2 \:=\:(90-x)^2 + y^2

    Differentiate with respect to time: . 2z\frac{dz}{dt} \:=\:-2(90-x)\frac{dx}{dt} + 2y\frac{dy}{dt}

    . . and we have: . \frac{dz}{dt} \;=\;\frac{1}{z}\left[(x-90)\frac{dx}{dt} + y\frac{dy}{dt}\right] .[1]


    When t = 1\!\;\;x = 22,\;y = 28,\;\frac{dx}{dt} = 22,\;\frac{dy}{dt} = 28

    . . and: . z \;=\;\sqrt{22^2+28^2} \:=\:\sqrt{1269} \;=\;3\sqrt{141}


    Substitute into [1]: . \frac{dz}{dt} \;=\;\frac{1}{3\sqrt{141}}\bigg[(22-90)(22) + (28)(28)\bigg] \;=\;\frac{-712}{3\sqrt{141}}


    Therefore: . \frac{dz}{dt} \;=\;-19.98707226 \;\approx\;\boxed{-20\text{ ft/sec}}\quad\hdots\quad\boxed{\text{decreasing}}

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Related Rates
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 26th 2009, 08:54 PM
  2. Rates and Related Rates!!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 2nd 2008, 10:53 AM
  3. Another Related Rates
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 13th 2008, 06:32 AM
  4. Related Rates
    Posted in the Calculus Forum
    Replies: 6
    Last Post: March 12th 2008, 05:49 PM
  5. rates and related rates
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 29th 2007, 09:51 PM

Search Tags


/mathhelpforum @mathhelpforum