Use the position equation given below, where s represents the height of the object (in feet), v0 represents the initial velocity of the object (in feet per second), s0 represents the initial height of the object (in feet), and t represents the time (in seconds), as the model for the problem.
s = −16t2 + v0t + s0
Some Major League Baseball pitchers can throw a fastball at speeds of up to and over 100 miles per hour. Assume a Major League Baseball pitcher throws a baseball straight up into the air at 100 miles per hour from a height of 6 feet 4 inches.
(a) Use the position equation to write a mathematical model for the height of the baseball.

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.

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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).

Use the position equation given below, where s represents the height of the object (in feet), v0 represents the initial velocity of the object (in feet per second), s0 represents the initial height of the object (in feet), and t represents the time (in seconds), as the model for the problem. s = −16t2 + v0t + s0. Some Major League Baseball pitchers can throw a fastball at speeds of up to and over 100 miles per hour. Assume a Major League Baseball pitcher throws a baseball straight up into the air at 100 miles per hour from a height of 6 feet 4 inches. (a) Use the position equation to write a mathematical model for the height of the baseball.

Explore how to model the height of a baseball thrown straight up by a Major League pitcher at 100 miles per hour using quadratic equations. Learn how to convert units, apply kinematic equations, and write a mathematical model to predict the baseball's height over time. Suitable for students in Grades 9-12.

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