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Let's break down the problem step by step and work towards the solution.

### Problem Summary:
- The baseball diamond is a square with **90 ft** on each side.
- A player is running from first base to second base at a rate of **14 ft/s**.
- We need to find the rate at which the player's distance from **home plate** is increasing when the player is halfway between first base and second base.

### Step 1: Geometry Setup
- Let the distance from **home plate** to the player be \( z \).
- Let the distance from **first base** to the player be \( x \).
  
From the diagram, the baseball diamond forms a right triangle between the home plate, the player, and the first base. By the **Pythagorean theorem**, we have:
\[
z^2 = x^2 + 90^2
\]
Where \( 90 \) ft is the distance from home plate to first base.

### Step 2: Differentiating with Respect to Time
We need to find the rate of change of \( z \) with respect to time \( t \), i.e., \( \frac{dz}{dt} \), when the player is halfway between first and second base (so \( x = 45 \) ft).

Implicitly differentiating the equation \( z^2 = x^2 + 90^2 \) with respect to time \( t \), we get:
\[
2z \frac{dz}{dt} = 2x \frac{dx}{dt}
\]
Simplifying:
\[
z \frac{dz}{dt} = x \frac{dx}{dt}
\]

### Step 3: Substituting Known Values
- When the player is halfway, \( x = 45 \) ft.
- \( \frac{dx}{dt} = 14 \) ft/s (this is the rate at which the player is running from first to second base).

Now, we need to find \( z \) when \( x = 45 \). Using the Pythagorean theorem:
\[
z^2 = 45^2 + 90^2
\]
\[
z^2 = 2025 + 8100 = 10125
\]
\[
z = \sqrt{10125} \approx 100.62 \, \text{ft}
\]

### Step 4: Solve for \( \frac{dz}{dt} \)
Now we can plug in the values for \( z \), \( x \), and \( \frac{dx}{dt} \) into the differentiated equation:
\[
100.62 \cdot \frac{dz}{dt} = 45 \cdot 14
\]
\[
100.62 \cdot \frac{dz}{dt} = 630
\]
\[
\frac{dz}{dt} = \frac{630}{100.62} \approx 6.26 \, \text{ft/s}
\]

### Conclusion:
It looks like the answer \( 6.26 \, \text{ft/s} \) was correct, but it was marked as incorrect in the system. There might be a rounding issue or an input issue in the answer field.

Would you like more clarification or further details?

Here are 5 related questions to expand your understanding:
1. How do related rates problems generally work in calculus?
2. What is the significance of implicit differentiation in solving related rates?
3. How would the problem change if the player ran faster or slower?
4. How can we use this concept to model real-world problems in physics or sports?
5. What happens to the rate of distance change if the player was running to third base instead of second?

**Tip:** Always ensure that units in the final answer match the required units in the problem to avoid unnecessary errors!

At what rate is the player's distance from home base increasing if he is halfway between first and second base, given that the baseball diamond is a square with 90 ft on each side and the player runs from first base to second base at 14 ft/s?

Solve a related rates problem where a player runs from first to second base on a square baseball diamond. Learn how to use the Pythagorean Theorem and implicit differentiation to calculate the rate at which the player's distance from home base is increasing. The player runs at 14 ft/s and is halfway between the bases.

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