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To solve for the speed of the baseball in miles per hour (mph) when it passes over home plate, let’s go through the details provided.

### Given Data
The baseball follows these parametric equations:
- \( x(t) = 144t \)
- \( y(t) = -16t^2 + 6t + 5 \)

Where:
- \( x'(t) = 144 \) ft/sec (horizontal velocity)
- \( y'(t) = -32t + 6 \) ft/sec (vertical velocity)

### Steps to Solve
1. **Calculate the Time, \( t \), When the Ball Reaches Home Plate:**
   - The distance to home plate is 60.5 feet.
   - Set \( x(t) = 60.5 \) and solve for \( t \):
     \[
     144t = 60.5
     \]
     \[
     t = \frac{60.5}{144}
     \]
     Calculating \( t \):
     \[
     t \approx 0.4201 \text{ seconds}
     \]

2. **Determine \( y'(t) \) at \( t = 0.4201 \):**
   - Substitute \( t \approx 0.4201 \) into \( y'(t) = -32t + 6 \):
     \[
     y'(0.4201) = -32(0.4201) + 6
     \]
     \[
     y'(0.4201) \approx -7.4432 \text{ ft/sec}
     \]

3. **Calculate the Speed \( s(t) \):**
   - The speed \( s(t) \) is given by:
     \[
     s(t) = \sqrt{[x'(t)]^2 + [y'(t)]^2}
     \]
   - Substitute \( x'(t) = 144 \) and \( y'(t) \approx -7.4432 \):
     \[
     s(0.4201) = \sqrt{(144)^2 + (-7.4432)^2}
     \]
     \[
     s(0.4201) \approx \sqrt{20736 + 55.42}
     \]
     \[
     s(0.4201) \approx \sqrt{20791.42}
     \]
     \[
     s(0.4201) \approx 144.2 \text{ ft/sec}
     \]

4. **Convert the Speed to Miles per Hour (mph):**
   - To convert from feet per second to miles per hour, use the conversion factor \( 1 \text{ ft/sec} = 0.681818 \text{ mph} \):
     \[
     s \approx 144.2 \times 0.681818
     \]
     \[
     s \approx 

A major league baseball pitcher throws a pitch that follows these parametric equations:

x(t) = 144t
y(t) = -16t^2 + 6t + 5

The time units are seconds, and the distance units are feet. The distance between the location of the pitcher and home plate (where the batter stands) is 60.5 feet. Give EXACT answers, unless instructed otherwise.

Calculate the horizontal velocity of the baseball at time t. What is the horizontal velocity of the baseball when it passes over home plate? What is the vertical velocity of the baseball at time t? Recall that the speed of the baseball at time t is given by s(t) = sqrt([x'(t)]^2 + [y'(t)]^2) ft/sec.

What is the speed of the baseball (in mph) when it passes over home plate?

This detailed solution covers how to calculate the speed of a baseball when it passes over home plate using parametric equations. Learn to compute horizontal and vertical velocities, and convert feet per second to miles per hour. Suitable for college-level physics and math students.

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