https://file.aiquickdraw.com/mathbot/file/17308688436878kfvw06w.jpg

The problem in the image involves parametric equations describing the motion of a baseball pitch, and it breaks down into several parts:

1. **Horizontal and Vertical Velocity**: 
   - The function \( x(t) = 144t \) has a horizontal velocity given by \( x'(t) = 144 \) ft/sec.
   - The function \( y(t) = -16t^2 + 6t + 5 \) has a vertical velocity given by \( y'(t) = -32t + 6 \) ft/sec.

2. **Speed Calculation**:
   - The speed \( s(t) \) at any time \( t \) is calculated as:
     \[
     s(t) = \sqrt{[x'(t)]^2 + [y'(t)]^2}
     \]
   - The speed over home plate (where \( x = 60.5 \) feet) was calculated by evaluating \( y'(t) \) at that point, then using this in the speed formula.

3. **Time to Hit the Ground**:
   - To determine the time when the baseball hits the ground, solve \( y(t) = 0 \), giving the quadratic equation solution shown.

4. **Angle of Impact**:
   - The angle at which the baseball hits the ground is calculated using the inverse tangent function, relating the vertical and horizontal velocities just before impact.

Would you like detailed solutions or explanations for any specific part? 

Here are some questions that might further enhance your understanding:

1. How do you calculate the speed of an object given its parametric equations for position?
2. Why does the horizontal velocity remain constant while the vertical velocity changes?
3. How is the angle of impact calculated from the final velocities?
4. What would change in the calculations if air resistance were considered?
5. How does the initial height affect the time it takes for the baseball to hit the ground?

**Tip**: When solving for the speed in parametric motion, always square and add the derivatives of each component, then take the square root to find the resultant speed.

A major league baseball pitcher throws a pitch that follows these parametric equations: x(t) = 144t and y(t) = -16t^2 + 6t + 5. The problem involves calculating the horizontal velocity, vertical velocity, speed at the home plate, time when the baseball hits the ground, and the angle of impact.

Solve a baseball pitch problem involving parametric equations for horizontal and vertical motion. Learn to calculate the speed, time of impact, and angle using derivatives, quadratic formula, and trigonometric functions. Suitable for calculus students.

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