To model the height of the baseball, we can use the given position equation:

$s(t) = -16t^2 + v_0t + s_0$

where:

• $s(t)$ is the height of the baseball at time $t$ (in feet),
• $v_0$ is the initial velocity of the baseball (in feet per second),
• $s_0$ is the initial height of the baseball (in feet),
• $t$ is the time (in seconds).

Step 1: Convert velocity from miles per hour to feet per second

The problem states that the baseball is thrown at 100 miles per hour. To convert this to feet per second, we can use the following conversion factors: $1 \text{ mile} = 5280 \text{ feet}, \quad 1 \text{ hour} = 3600 \text{ seconds}$ Thus, $100 \, \text{miles/hour} = 100 \times \frac{5280}{3600} \, \text{feet/second} = 146.67 \, \text{feet/second}$

So the initial velocity $v_0 = 146.67 \, \text{ft/sec}$.

Step 2: Convert the initial height to feet

The initial height is given as 6 feet 4 inches. Since there are 12 inches in a foot, we can convert 4 inches to feet: $4 \, \text{inches} = \frac{4}{12} = 0.3333 \, \text{feet}$ So the initial height $s_0 = 6 + 0.3333 = 6.3333 \, \text{feet}$.

Step 3: Write the mathematical model

Now that we have the initial velocity $v_0 = 146.67 \, \text{ft/sec}$ and the initial height $s_0 = 6.3333 \, \text{feet}$, we can substitute these values into the position equation:

$s(t) = -16t^2 + 146.67t + 6.3333$

This is the mathematical model for the height of the baseball over time.

---

Would you like further details or have any questions?

Here are some follow-up questions for you:

1. How would you determine the maximum height the baseball reaches?
2. How long does it take for the baseball to reach its maximum height?
3. How long does it take for the baseball to hit the ground?
4. What is the velocity of the baseball at its maximum height?
5. Can you graph the height of the baseball as a function of time?

Tip: When converting units, double-check conversion factors to ensure accuracy, especially when switching between systems (e.g., miles to feet, hours to seconds).