A ball is launched at 42 feet per second. The height of the ball (in feet) is given by the equation h=-16t^2+48t+6, where t is the time (in seconds) since the ball was launched. For how many seconds was the ball at or above a height of 38 feet?

To find the time the ball is at or above a height of 38 feet, we start with the equation for the height of the ball:

\[
h = -16t^2 + 48t + 6
\]

We need to set this equation greater than or equal to 38:

\[
-16t^2 + 48t + 6 \geq 38
\]

Rearranging gives:

\[
-16t^2 + 48t + 6 - 38 \geq 0
\]

\[
-16t^2 + 48t - 32 \geq 0
\]

Dividing the entire inequality by -16 (remembering to flip the inequality sign):

\[
t^2 - 3t + 2 \leq 0
\]

Next, we factor the quadratic:

\[
(t - 1)(t - 2) \leq 0
\]

Now, we determine the intervals where this inequality holds. The roots are \( t = 1 \) and \( t = 2 \). We test the intervals around these points:

1. **Interval \( (-\infty, 1) \)**: Choose \( t = 0 \):
   \[
   (0 - 1)(0 - 2) = 2 > 0
   \]

2. **Interval \( (1, 2) \)**: Choose \( t = 1.5 \):
   \[
   (1.5 - 1)(1.5 - 2) = (-0.25) < 0
   \]

3. **Interval \( (2, \infty) \)**: Choose \( t = 3 \):
   \[
   (3 - 1)(3 - 2) = 2 > 0
   \]

The inequality \( (t - 1)(t - 2) \leq 0 \) holds for:

\[
1 \leq t \leq 2
\]

Thus, the ball is at or above a height of 38 feet from \( t = 1 \) second to \( t = 2 \) seconds.

### Duration
The duration for which the ball is at or above 38 feet is:

\[
2 - 1 = 1 \text{ second}
\]

### Summary
The ball is at or above a height of 38 feet for **1 second**.

Would you like further details or explanations? Here are some related questions:

1. What are the maximum height and time at which it occurs?
2. How does the ball's height change after 2 seconds?
3. What would happen if the launch speed were increased?
4. How would you graph the height function?
5. What is the total time the ball is in the air? 

**Tip**: Always check the endpoints of your intervals when solving inequalities.

Discover how to determine the time a ball remains at or above a height of 38 feet using the quadratic equation h = -16t^2 + 48t + 6. This solution involves algebraic manipulation and inequality analysis, suitable for Grades 8-10.

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