To solve the inequality:
[tex]\[
-4.9 t^2 + 22 t + 0.75 > 10
\][/tex]
we need to follow these steps:
1. Simplify the inequality:
Subtract 10 from both sides to set the inequality to zero:
[tex]\[
-4.9 t^2 + 22 t + 0.75 - 10 > 0
\][/tex]
Simplify further:
[tex]\[
-4.9 t^2 + 22 t - 9.25 > 0
\][/tex]
2. Find the roots of the corresponding quadratic equation:
Solve the equation:
[tex]\[
-4.9 t^2 + 22 t - 9.25 = 0
\][/tex]
We can use the quadratic formula for this:
[tex]\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = -4.9 \)[/tex], [tex]\( b = 22 \)[/tex], and [tex]\( c = -9.25 \)[/tex].
Calculate the discriminant:
[tex]\[
b^2 - 4ac = 22^2 - 4(-4.9)(-9.25) = 484 - 4(4.9)(9.25)
\][/tex]
Simplify the discriminant:
[tex]\[
b^2 - 4ac = 484 - 181.3 = 302.7
\][/tex]
Take the square root of the discriminant:
[tex]\[
\sqrt{302.7} \approx 17.4
\][/tex]
Now, substitute back into the quadratic formula:
[tex]\[
t = \frac{-22 \pm 17.4}{-9.8}
\][/tex]
Find the two solutions:
[tex]\[
t_1 = \frac{22 - 17.4}{9.8} \approx 0.469
\][/tex]
[tex]\[
t_2 = \frac{22 + 17.4}{9.8} \approx 4.02
\][/tex]
3. Determine the intervals where the inequality holds:
The parabola [tex]\( -4.9 t^2 + 22 t - 9.25 \)[/tex] opens downward (because the coefficient of [tex]\( t^2 \)[/tex] is negative). Therefore, the height will be greater than 10 meters between the two roots.
Thus, the ball’s height is greater than 10 meters when [tex]\( t \)[/tex] is approximately between [tex]\( 0.469 \)[/tex] and [tex]\( 4.02 \)[/tex] seconds.
The correct answer, when the ball's height is greater than 10 meters, is:
[tex]\[
\boxed{0.469 \text{ and } 4.02}
\][/tex]
