A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 24 feet per second is 22 feet from third base. At what rate (in ft/sec) is the player's distance from home plate changing? (Round your answer to two decimal places.)

To determine the rate at which the player's distance from home plate is changing, we need to apply the concept of related rates in calculus.

Let's define the variables involved:
- Let \( D \) be the distance from the player to home plate.
- Let \( x \) be the distance from the player to third base.
- Let \( y \) be the distance from the player to home plate along the path from third base to home plate (hypotenuse of the right triangle formed).

Given:
- The side of the square (base paths) is 90 feet.
- The player is running from second base to third base at a speed of 24 feet per second.
- The player is 22 feet from third base.

### Step-by-Step Solution

1. **Determine \( y \):**
   The distance \( y \) can be found using the Pythagorean theorem. The distances from home plate to third base and third base to the player form a right triangle:
   \[
   y = \sqrt{90^2 + x^2}
   \]
   Given \( x = 22 \) feet (distance from the player to third base):
   \[
   y = \sqrt{90^2 + 22^2} = \sqrt{8100 + 484} = \sqrt{8584} \approx 92.63 \text{ feet}
   \]

2. **Differentiate with respect to time \( t \):**
   We need the rate at which \( y \) is changing (\(\frac{dy}{dt}\)). To find this, we differentiate both sides of the Pythagorean theorem with respect to time \( t \):
   \[
   \frac{d}{dt}(y^2) = \frac{d}{dt}(90^2 + x^2)
   \]
   Using the chain rule:
   \[
   2y \frac{dy}{dt} = 2x \frac{dx}{dt}
   \]

3. **Solve for \(\frac{dy}{dt}\):**
   \[
   y \frac{dy}{dt} = x \frac{dx}{dt}
   \]
   We know \( \frac{dx}{dt} = -24 \) feet per second (negative because the distance to third base is decreasing):
   \[
   92.63 \frac{dy}{dt} = 22 \times (-24)
   \]
   \[
   92.63 \frac{dy}{dt} = -528
   \]
   \[
   \frac{dy}{dt} = \frac{-528}{92.63} \approx -5.70 \text{ feet per second}
   \]

### Conclusion

The rate at which the player's distance from home plate is changing is approximately \(-5.70\) feet per second. (The negative sign indicates the distance is decreasing.)

Would you like more details or have any questions?

Here are some additional questions you might want to explore:
1. What is the distance between second base and home plate when the player is 22 feet from third base?
2. How long does it take for the player to reach third base from their current position?
3. At what rate is the player's distance from first base changing when they are 22 feet from third base?
4. How would the rate change if the player were running towards home plate instead of third base?
5. What would be the player's distance from home plate if they were running at a different speed?

**Tip:** When solving related rates problems, always start by identifying and clearly defining the variables and their relationships using geometric or physical principles.

Explore a related rates problem involving a baseball diamond where a player runs from second base to third base. Understand how to apply calculus concepts, the Pythagorean theorem, and the chain rule to determine the rate of change of the player's distance from home plate. Get a detailed solution with step-by-step calculations.

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