In a baseball game, a ball is hit down the first baseline, as shown here.

As soon as the batter hits the ball, a base runner on third base starts running towards home plate.

The first baseman fields the ball and throws it towards home plate.

As the base runner and ball move towards home plate, the distance between them changes.

If we wish to determine the rate at which the distance between the base runner and the ball changes, measuring it directly is not a viable option. Let's examine an alternative.

We can make use of the fact that the rate of change of the distance between the runner and the ball is related to the rate of change of the distance between home plate and the ball.

This rate of change is also related to the rate of change of the distance between home plate and the base runner.

Both of these rates are easier to measure than the rate of change of the distance between the base runner and the ball. Perhaps, then, it is possible to compute the rate of change of the distance between the base runner and ball in terms of the other two rates.

This figure illustrates the basic idea of a related rates problem: One rate of change is computed in terms of one or more other rates of change. Let's use this approach to set up and solve our relate-rates baseball problem.

The baseball diamond is actually a square, with each side 90 feet long. Let's rotate the diamond 45° so that the first baseline is oriented in the x direction, and the third baseline is along the y direction.

The first baseman fields the ball 24 feet from home plate and throws the ball towards home plate at 100 feet per second. We'll call the time at which the first baseman fields the ball t = 0.

If we connect the positions of the base runner, the ball, and home plate, we see that the connecting lines form a right triangle. Let's take the variable z to be the distance between the base runner and ball.

How is the distance z related to the distances x and y? Click the "Submit" button after selecting your answer.

The Pythagorean theorem shows how the three distances x, y, and z are related.

Use the slider to vary the time t. See how the distances x, y, and z change with respect to time.

Which distance shows the greatest rate of change? Click the "Submit" button after selecting your answer.

We wish to determine the rate of change of z. To do this we first use the chain rule to differentiate both sides of the Pythagorean theorem equation.

We then substitute into the equation the values of the rates of change in x and y. Both of these rates are negative, because both distances are decreasing.

Now, let's determine the rate of change of z at t = 0.2 seconds.

At time t = 0.2 seconds, the base runner has advanced 4 feet towards home plate, and the ball has traveled 20 feet. We can use these values to determine x and y in our equation.

We then use the Pythagorean theorem to determine the value of z at that time.

Finally, we substitute the values into the equation and solve for the rate of change in z.

When the ball is thrown, the base runner is 7 feet from home plate and is moving at 20 feet per second,

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