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I see the math problem in the image involves analyzing the motion of a baseball through parametric equations, covering several steps:

1. **Calculating horizontal and vertical velocities** using derivatives of the given parametric equations.
2. **Finding the speed of the baseball** by combining the horizontal and vertical velocities at the point it passes over home plate.
3. **Determining when the baseball hits the ground** by solving for the time \( t \) when the height function \( y(t) \) becomes zero.
4. **Calculating the angle of impact** in radians, which involves finding the angle of the velocity vector at the moment the baseball hits the ground.

Here is an outline of the solutions based on the visible parts:

- **Horizontal Velocity (x'(t))**: It's provided directly as \( 144 \) ft/sec.
- **Vertical Velocity (y'(t))**: Calculated as \( -32t + 6 \).
- **Speed Calculation**: Uses \( s(t) = \sqrt{[x'(t)]^2 + [y'(t)]^2} \).
- **Time to Hit Ground**: Involves solving \( y(t) = 0 \).
- **Angle of Impact**: Calculated using the angle between the tangent line and the path of the ball, based on the components of velocity.

Would you like me to go through each of these steps in detail or help with a specific part? Here are five related questions to expand on these concepts:

1. How do we interpret the direction of the angle in projectile motion?
2. What would change in the calculations if air resistance was a factor?
3. How can parametric equations like these be applied to other sports or motion analysis scenarios?
4. What is the significance of converting the speed from ft/sec to mph in this context?
5. How would we solve this problem if only approximate answers were allowed?

**Tip:** Remember that breaking down complex motion problems into smaller parts, like analyzing horizontal and vertical components separately, simplifies solving.

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.

(a) Calculate the horizontal velocity of the baseball at time t.
(b) What is the horizontal velocity of the baseball when it passes over home plate?
(c) What is the vertical velocity of the baseball at time t.
(d) What is the speed of the baseball (in mph) when it passes over home plate?
(e) At what time does the baseball hit the ground, assuming the batter and catcher miss the ball?
(f) What is the magnitude of the angle at which the baseball hits the ground?

Explore how to calculate the speed and angle of impact of a baseball pitch using parametric equations. Learn how to find horizontal and vertical velocities, determine speed at home plate, calculate the time of impact, and find the angle of impact when the baseball hits the ground. This step-by-step solution applies calculus and physics concepts, ideal for college-level students.

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