To solve the given inequality and determine the time interval when the ball's height is greater than 10 meters, we start with the provided inequality:
[tex]\[ -4.9t^2 + 22t + 0.75 > 10 \][/tex]
First, we rearrange the inequality to bring all the terms to one side:
[tex]\[ -4.9t^2 + 22t + 0.75 - 10 > 0 \][/tex]
[tex]\[ -4.9t^2 + 22t - 9.25 > 0 \][/tex]
Next, we solve for [tex]\( t \)[/tex] by finding the roots of the corresponding equation:
[tex]\[ -4.9t^2 + 22t - 9.25 = 0 \][/tex]
The roots of this quadratic equation represent the times when the ball’s height is exactly 10 meters. These can be found using the quadratic formula:
[tex]\[ t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Where:
- [tex]\( a = -4.9 \)[/tex]
- [tex]\( b = 22 \)[/tex]
- [tex]\( c = -9.25 \)[/tex]
The discriminant is:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 22^2 - 4(-4.9)(-9.25) \][/tex]
Solving for [tex]\( \Delta \)[/tex] yields two real roots, which are the solutions to the quadratic equation. These roots define the interval within which the ball's height is exactly 10 meters. To determine the interval where the height is greater than 10 meters, we test the inequality in the intervals around these roots.
Finally, the interval of time [tex]\( t \)[/tex] during which the ball's height is greater than 10 meters is:
[tex]\[ \boxed{0.469563696261278 \text{ and } 4.02023222210607} \][/tex]
Thus, the ball's height is greater than 10 meters when [tex]\( t \)[/tex] is approximately between 0.469563696261278 and 4.02023222210607 seconds.
