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.

To determine whether the baseball travels over the fence, we need to evaluate the height of the baseball when it reaches the horizontal distance of 320 feet. 

### Step 1: Find the time \( t \) when the baseball reaches the fence
The horizontal position \( x \) of the baseball is given by:

\[
x = 140t
\]

We want to find the time \( t \) when the baseball reaches the fence, which is 320 feet away. Set \( x = 320 \) and solve for \( t \):

\[
320 = 140t
\]

\[
t = \frac{320}{140} = \frac{32}{14} \approx 2.29 \, \text{seconds}
\]

### Step 2: Find the height \( y \) of the baseball at that time
The vertical position \( y \) of the baseball is given by:

\[
y = 3.1 + 40t - 16t^2
\]

Substitute \( t = 2.29 \) into the equation:

\[
y = 3.1 + 40(2.29) - 16(2.29)^2
\]

Calculate each term:

\[
y \approx 3.1 + 91.6 - 16(5.2441)
\]

\[
y \approx 3.1 + 91.6 - 83.9056
\]

\[
y \approx 10.7944 \, \text{feet}
\]

### Step 3: Compare the height of the baseball with the height of the fence
The fence is 10 feet tall. The height of the baseball at the point where it reaches the fence is approximately 10.79 feet.

### Conclusion
Since 10.79 feet is greater than the 10-foot height of the fence, the baseball **does** travel over the fence.

---

Would you like further details or have any questions? 

Here are 5 related questions to expand your understanding:

1. How would the answer change if the fence were located at a different distance from home plate?
2. What would the maximum height of the baseball be, and at what time does it occur?
3. If the initial height of the baseball were different, how would that affect whether it clears the fence?
4. How does the coefficient of the \( t^2 \) term affect the baseball's trajectory?
5. What would happen if the baseball were hit with a different initial velocity?

**Tip:** Understanding how to manipulate quadratic equations is crucial when dealing with projectile motion, as they often describe the vertical displacement in physics problems.

Explore how to determine if a baseball clears a 10-foot fence using projectile motion equations and quadratic equations. Detailed step-by-step solution provided. Suitable for advanced high school students.

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