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