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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.

  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.

  3. Write a formula that involves the variables:


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

  4. Differentiate with respect to time:

  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:


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


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

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