A ball is thrown from an initial height of 6 feet with an initial upward velocity of 33 ft./s the balls height H in feet after T seconds is given the following. Find all the values of tea for which the balls height is 22 feet round your answers to the nearest hundredth.

The height of the ball as a function of time \( T \) is given by the equation:

\[
H(T) = -16T^2 + 33T + 6
\]

You want to find the times \( T \) for which the height \( H(T) \) is 22 feet. So, we set \( H(T) = 22 \) and solve the quadratic equation:

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

First, subtract 22 from both sides:

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

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

Now, solve this quadratic equation using the quadratic formula:

\[
T = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our equation, \( a = -16 \), \( b = 33 \), and \( c = -16 \). Plugging in these values:

\[
T = \frac{-33 \pm \sqrt{33^2 - 4(-16)(-16)}}{2(-16)}
\]

\[
T = \frac{-33 \pm \sqrt{1089 - 1024}}{-32}
\]

\[
T = \frac{-33 \pm \sqrt{65}}{-32}
\]

\[
T = \frac{-33 \pm 8.06}{-32}
\]

This gives two solutions:

\[
T_1 = \frac{-33 + 8.06}{-32} \approx \frac{-24.94}{-32} \approx 0.78
\]

\[
T_2 = \frac{-33 - 8.06}{-32} \approx \frac{-41.06}{-32} \approx 1.28
\]

So, the ball is at a height of 22 feet at approximately \( T = 0.78 \) seconds and \( T = 1.28 \) seconds.

Would you like a more detailed explanation or have any questions? Here are 5 related questions for practice:

1. What is the maximum height the ball reaches?
2. At what time does the ball hit the ground?
3. How long does it take for the ball to reach its maximum height?
4. How would the initial velocity affect the time the ball reaches a specific height?
5. How would the initial height change the overall trajectory?

**Tip:** For any quadratic equation, the vertex gives important information. In projectile motion, the vertex represents the peak of the object's path.

Learn how to solve the projectile motion problem where a ball is thrown with initial height and velocity, finding the time when its height reaches 22 feet. Detailed step-by-step solution provided.

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