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 26 feet per second is 25 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.)


We can solve this problem using related rates in calculus. Here's the step-by-step solution:

### Step 1: Understanding the Geometry
The baseball diamond is a square with 90-foot sides, and the player is running from second base to third base. The distance from second base to third base is 90 feet.

- Let \( x \) be the distance between the player and third base.
- Let \( z \) be the distance between the player and home plate.
- The player's speed is given as \( \frac{dx}{dt} = -26 \) feet per second (negative because the distance to third base is decreasing).

At the moment we are interested in, the player is 25 feet from third base, so \( x = 25 \) feet.

### Step 2: Applying the Pythagorean Theorem
Since the player is running on the baseline, and the distance from third base to home plate is also 90 feet, the distance between the player and home plate, \( z \), forms the hypotenuse of a right triangle with legs \( x \) and 90 feet.

By the Pythagorean theorem:

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

### Step 3: Differentiate with Respect to Time
Now, differentiate both sides of the equation with respect to time \( t \):

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

We can cancel the 2's:

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

### Step 4: Substitute Known Values
We need to find \( z \) and then solve for \( \frac{dz}{dt} \).

1. First, calculate \( z \) when \( x = 25 \):

\[
z^2 = 25^2 + 90^2
\]
\[
z^2 = 625 + 8100 = 8725
\]
\[
z = \sqrt{8725} \approx 93.38 \text{ feet}
\]

2. Now, substitute the values \( z = 93.38 \), \( x = 25 \), and \( \frac{dx}{dt} = -26 \) into the equation:

\[
93.38 \cdot \frac{dz}{dt} = 25 \cdot (-26)
\]
\[
93.38 \cdot \frac{dz}{dt} = -650
\]

3. Solve for \( \frac{dz}{dt} \):

\[
\frac{dz}{dt} = \frac{-650}{93.38} \approx -6.96 \text{ feet per second}
\]

### Final Answer:
The player's distance from home plate is decreasing at approximately **6.96 feet per second**.

Would you like a more detailed explanation on any part of this solution? Here are some related questions for further exploration:

1. How does the Pythagorean theorem help in related rates problems?
2. Why is the player's speed negative in the equation?
3. What happens if the player was running towards first base instead?
4. How would the rate of change differ if the player was further from third base?
5. Can this problem be generalized to any square-shaped field?

**Tip:** When solving related rates problems, always identify the variables and their rates of change before setting up an equation!

A baseball diamond has the shape of a square with sides 90 feet long. A player running from second base to third base at a speed of 26 feet per second is 25 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.)

Solve a related rates calculus problem involving a baseball player running from second to third base at a speed of 26 ft/sec. Using the Pythagorean theorem and differentiation, learn how to find the rate at which the player's distance from home plate is changing. Suitable for advanced calculus students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation