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How to Do a Related Rate Problem Involving a Moving Baseball

You can use calculus to determine a rate that’s related to the speed of a moving object. For example, say a pitcher delivers a fastball, which the batter pops up — it goes straight up above home plate. When it reaches a height of 60 feet, it’s moving up at a rate of 50 feet per second. At this point, how fast is the distance from the ball to second base growing? Note: The distance between the bases of a baseball diamond is 90 feet.

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  1. Draw your diagram and label it as shown in the figure.

  2. List all given rates and the rate you’re asked to figure out.

    image1.png
  3. Write a formula that involves the variables:

    image2.png

    The Pythagorean Theorem is frequently used in related rate problems that involve a right triangle.

  4. Differentiate with respect to time:

    image3.png
  5. Substitute known values into this equation and solve for dd/dt:

    You’re missing a needed value, d. So use the Pythagorean Theorem again to get it:

    image4.png

    (You can reject the negative answer.) Now do the substitutions:

    image5.png

So the distance is growing at a rate of approximately 21.3 feet/second.

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