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### Problem 108:
**Question:** The height \(h\) (in meters) of a kangaroo’s jump is modeled by \(h(t) = -4.9t^2 + 9.8t\), where \(t\) is time in seconds. Determine the interval on which the kangaroo's height is at least 5 meters.

**Solution:**
To find the interval where the kangaroo's height is at least 5 meters, solve the inequality:

\[
h(t) = -4.9t^2 + 9.8t \geq 5
\]

Rearrange the inequality:

\[
-4.9t^2 + 9.8t - 5 \geq 0
\]

To solve the quadratic inequality, we first solve the corresponding quadratic equation:

\[
-4.9t^2 + 9.8t - 5 = 0
\]

Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -4.9 \), \( b = 9.8 \), and \( c = -5 \):

\[
t = \frac{-9.8 \pm \sqrt{(9.8)^2 - 4(-4.9)(-5)}}{2(-4.9)}
\]

\[
t = \frac{-9.8 \pm \sqrt{96.04 - 98}}{-9.8}
\]

\[
t = \frac{-9.8 \pm \sqrt{-1.96}}{-9.8}
\]

The discriminant is negative, indicating that the quadratic equation has no real roots. However, since the height function represents a parabola opening downward, the height will be greater than 5 meters for some interval of \( t \).

We should check at what times the height is exactly 5 meters by evaluating the inequality:

\[
h(t) = -4

Explore how to determine the time interval during which a kangaroo's jump height is at least 5 meters using the quadratic function model. This problem involves solving a quadratic inequality and applying the quadratic formula. Suitable for high school students studying quadratic functions and inequalities.

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