$$ h=6+33t-16t $$$$  $$$$ ^2 $$$$  $$$$  $$

It seems like you're trying to express the equation for the ball's height more clearly. Based on what you provided, the correct mathematical expression should be:

\[
h = 6 + 33t - 16t^2
\]

This equation represents the height $ h $ of the ball at time $ t $ seconds. You want to find when $ h = 22 $ feet. So we set the equation equal to 22 and solve for $ t $:

\[
6 + 33t - 16t^2 = 22
\]

First, subtract 22 from both sides:

\[
6 + 33t - 16t^2 - 22 = 0
\]

\[
-16t^2 + 33t - 16 = 0
\]

This is the same quadratic equation as before:

\[
-16t^2 + 33t - 16 = 0
\]

We already solved this equation using the quadratic formula and found the solutions:

\[
t \approx 0.78 \text{ seconds and } t \approx 1.28 \text{ seconds}
\]

So, the ball is at a height of 22 feet at approximately $ t = 0.78 $ seconds and $ t = 1.28 $ seconds.

Would you like to see more details or have any other questions? Here are 5 related questions you might explore:

1. What are the initial velocity and initial height in this context?
2. How do we determine when the ball reaches its maximum height?
3. What is the significance of the coefficients in a quadratic equation for projectile motion?
4. How would the equation change if the ball were thrown downward instead of upward?
5. How do air resistance and gravity influence the motion of a projectile in this equation?

**Tip:** When solving quadratic equations, always check if the discriminant $ b^2 - 4ac $ is positive, zero, or negative. This tells you how many real solutions there are.

Explore how to solve projectile motion equations to find the time when the height of a ball reaches 22 feet. This problem involves quadratic equations and is suitable for advanced high school students.

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