At what rate is his distance from third base changing at the same moment? ( **Hint** : At time t, let x denote the distance of the batter from the home plate, and z denote the distance of the batter from the third base.)
Answer =

To determine how fast the distance from third base (denoted as \( z \)) is changing, we need to apply related rates and use the Pythagorean Theorem. Let’s break it down step by step:

### Step 1: Understanding the Setup

In a standard baseball diamond, the bases form a square with 90-foot sides. The batter runs from home plate towards first base, and we want to determine how fast his distance from third base is changing at a specific moment.

- Let \( x \) represent the distance of the batter from home plate at any time \( t \).
- Let \( z \) represent the distance of the batter from third base at any time \( t \).
- The distance between home plate and third base is the fixed side of the baseball diamond, which is 90 feet.

### Step 2: Relating \( x \) and \( z \)

By the Pythagorean Theorem, the relationship between \( x \), \( z \), and the fixed distance from home plate to third base (90 feet) is:

\[
z^2 = x^2 + 90^2
\]

### Step 3: Differentiating with Respect to Time \( t \)

To find how fast \( z \) is changing with respect to time, differentiate both sides of the equation with respect to \( t \):

\[
2z \frac{dz}{dt} = 2x \frac{dx}{dt}
\]

This simplifies to:

\[
z \frac{dz}{dt} = x \frac{dx}{dt}
\]

Now, we need to solve for \( \frac{dz}{dt} \), the rate at which the distance from third base is changing:

\[
\frac{dz}{dt} = \frac{x}{z} \cdot \frac{dx}{dt}
\]

### Step 4: Using Given Information

- \( \frac{dx}{dt} \) is the speed at which the batter is running towards first base (given in the problem).
- \( x \) is the current distance of the batter from home plate (given in the problem).
- \( z \) is the distance from third base, which can be computed from the Pythagorean relationship:
  \[
  z = \sqrt{x^2 + 90^2}
  \]

### Step 5: Substitute and Calculate

You can now substitute the values for \( x \), \( \frac{dx}{dt} \), and \( z \) (computed from the above equation) to find \( \frac{dz}{dt} \), the rate at which the distance from third base is changing.

If you provide specific values for the speed of the batter or distance, I can give you the final numeric solution.

Would you like me to go over any part of the solution in more detail? Here are some follow-up questions:

1. How does the Pythagorean theorem relate to this problem?
2. What does \( \frac{dx}{dt} \) represent physically in this context?
3. What additional information would you need to solve for \( \frac{dz}{dt} \) numerically?
4. How does \( \frac{dz}{dt} \) change as the batter gets closer to first base?
5. How can we visualize the rates of change in terms of the baseball diamond?

**Tip**: When solving related rates problems, always differentiate after establishing relationships between variables through a geometric or algebraic formula.

At what rate is his distance from third base changing at the same moment? (Hint: At time t, let x denote the distance of the batter from the home plate, and z denote the distance of the batter from the third base.)

Explore how to solve a related rates problem involving a batter running towards first base and the changing distance from third base. This solution uses the Pythagorean Theorem and differentiation to find the rate of change of distance at a specific moment. Suitable for calculus students in Grades 11-12 or early college.

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