To determine the time intervals during which the ball's height is greater than 10 meters, we need to solve the given inequality:
[tex]\[
-4.9 t^2 + 22 t + 0.75 > 10
\][/tex]
First, let's simplify the inequality by moving the constant term from the right-hand side to the left-hand side:
[tex]\[
-4.9 t^2 + 22 t + 0.75 - 10 > 0
\][/tex]
This simplifies to:
[tex]\[
-4.9 t^2 + 22 t - 9.25 > 0
\][/tex]
Next, we need to find the values of [tex]\( t \)[/tex] that satisfy this inequality. Solving this quadratic inequality involves finding the roots of the corresponding quadratic equation:
[tex]\[
-4.9 t^2 + 22 t - 9.25 = 0
\][/tex]
Once the roots of this equation are determined, we can identify the intervals where the quadratic expression is greater than zero.
The roots of this quadratic equation are approximately:
[tex]\[
t_1 \approx 0.469563696261278
\][/tex]
[tex]\[
t_2 \approx 4.02023222210607
\][/tex]
Since the quadratic coefficient is negative (-4.9), the parabola opens downwards. Therefore, the quadratic expression is greater than zero between the roots. So, the ball's height is greater than 10 meters in the interval:
[tex]\[
0.469563696261278 < t < 4.02023222210607
\][/tex]
Thus, the ball's height is greater than 10 meters when [tex]\( t \)[/tex] is approximately between [tex]\( 0.469563696261278 \)[/tex] and [tex]\( 4.02023222210607 \)[/tex] seconds.
In summary:
The ball's height is greater than 10 meters when [tex]\( t \)[/tex] is approximately between [tex]\(\boxed{0.469563696261278}\)[/tex] and [tex]\(\boxed{4.02023222210607}\)[/tex] seconds.
