A baseball is hit, following a path represented by x=140t and y=3.1+40t-16t to the power of 2 for 0 is less than greater to t less than greater to 3. The fence, which is 10 feet tall, lies 320 feet away from home plate. Does the baseball travel over the fence? Justify your answer mathematically. Write  a rectangular equation to represent the plane curve.

To determine whether the baseball travels over the fence and to write a rectangular equation for the plane curve, follow these steps:

### 1. **Given Parametric Equations:**

- \( x = 140t \)
- \( y = 3.1 + 40t - 16t^2 \)

### 2. **Convert the Parametric Equations into a Rectangular Equation:**

We want to eliminate the parameter \( t \) to obtain a rectangular equation in terms of \( x \) and \( y \).

From the first equation:
\[
x = 140t
\]
Solve for \( t \):
\[
t = \frac{x}{140}
\]

Substitute this expression for \( t \) into the equation for \( y \):

\[
y = 3.1 + 40\left(\frac{x}{140}\right) - 16\left(\frac{x}{140}\right)^2
\]

Simplify the equation:

\[
y = 3.1 + \frac{40x}{140} - 16\left(\frac{x^2}{19600}\right)
\]

\[
y = 3.1 + \frac{2x}{7} - \frac{16x^2}{19600}
\]

Further simplifying:

\[
y = 3.1 + \frac{2x}{7} - \frac{x^2}{1225}
\]

So, the rectangular equation representing the plane curve is:
\[
y = 3.1 + \frac{2x}{7} - \frac{x^2}{1225}
\]

### 3. **Determine Whether the Baseball Clears the Fence:**

The fence is located 320 feet away from home plate, so substitute \( x = 320 \) into the rectangular equation to find the height \( y \) of the baseball at that point:

\[
y = 3.1 + \frac{2(320)}{7} - \frac{(320)^2}{1225}
\]

Calculate each term:

\[
y = 3.1 + \frac{640}{7} - \frac{102400}{1225}
\]

\[
y \approx 3.1 + 91.43 - 83.59
\]

\[
y \approx 10.94 \, \text{feet}
\]

### 4. **Conclusion:**

Since the height of the baseball at 320 feet is approximately 10.94 feet, which is greater than the 10-foot height of the fence, the baseball does indeed travel over the fence.

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Would you like further details or have any questions? 

Here are 5 related questions to explore further:

1. How can you determine the maximum height of the baseball using the rectangular equation?
2. What happens if the initial speed of the baseball increases? How does that affect the trajectory?
3. How would you find the point at which the baseball reaches its maximum height?
4. How does changing the distance of the fence impact the equation used to determine if the ball clears the fence?
5. How would the equation change if the initial height of the baseball were different?

**Tip:** Converting parametric equations to rectangular form is a valuable skill, especially in physics and engineering, where analyzing trajectories and paths is common.

Explore whether a baseball clears a 10-foot fence using parametric equations and quadratic functions. This detailed solution involves converting parametric equations to rectangular form, calculating specific points on the trajectory, and determining the baseball's path over a 320-foot distance.

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