1. **Stating the problem:** We want to estimate the distance a ball will travel when hit with a speed of 107 mph. 2. **Relevant formula:** The distance a projectile travels depends on its initial speed, launch angle, and acceleration due to gravity. A common simplified formula for the range $R$ of a projectile launched at speed $v$ and angle $\theta$ is: $$R = \frac{v^2 \sin(2\theta)}{g}$$ where $g \approx 9.8$ m/s$^2$ is the acceleration due to gravity. 3. **Important notes:** - The maximum range occurs at $\theta = 45^\circ$. - We need to convert speed from mph to m/s for consistent units. 4. **Convert speed:** $$107 \text{ mph} = 107 \times 0.44704 = 47.85 \text{ m/s}$$ 5. **Calculate maximum range:** At $\theta = 45^\circ$, $\sin(2\theta) = \sin(90^\circ) = 1$, so $$R = \frac{(47.85)^2 \times 1}{9.8} = \frac{2290.82}{9.8} = 233.75 \text{ meters}$$ 6. **Interpretation:** The ball is most likely to travel approximately 234 meters if hit at an optimal angle of 45 degrees with a speed of 107 mph. **Final answer:** $$\boxed{234 \text{ meters}}$$