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The problem involves determining the interval of time during which a rock remains in the air after being thrown from a 124-ft-tall building, with its height described by the quadratic equation:

\[
h(t) = -16t^2 - 30t + 124
\]

Where:
- \( h(t) \) is the height of the rock at time \( t \) (in seconds),
- \( t \) is the time in seconds.

### To solve this, we need to find when the rock reaches the ground:
The rock will hit the ground when \( h(t) = 0 \). So, we need to solve the equation:

\[
-16t^2 - 30t + 124 = 0
\]

This is a quadratic equation, and we can solve it using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( -16t^2 - 30t + 124 = 0 \), the coefficients are:
- \( a = -16 \)
- \( b = -30 \)
- \( c = 124 \)

#### Step 1: Calculate the discriminant
\[
\Delta = b^2 - 4ac = (-30)^2 - 4(-16)(124) = 900 + 7936 = 8836
\]

#### Step 2: Apply the quadratic formula
\[
t = \frac{-(-30) \pm \sqrt{8836}}{2(-16)} = \frac{30 \pm 94}{-32}
\]

Now, solve for both possible values of \( t \):

1. \( t_1 = \frac{30 + 94}{-32} = \frac{124}{-32} = -3.875 \)
2. \( t_2 = \frac{30 - 94}{-32} = \frac{-64}{-32} = 2 \)

### Conclusion:
The rock is in the air from \( t = 0 \) (when it's dropped) until \( t = 2 \) seconds (when it hits the ground). Negative time is not physically meaningful in this context, so we discard \( t = -3.875 \).

The correct interval of time is \( 0 < t \leq 2 \).

Thus, the correct option is:
\[
\boxed{0 < t \leq 2}
\]

Would you like more details on how the quadratic formula was applied or further clarification?

### Related questions:
1. How do you determine when an object hits the ground in a projectile motion problem?
2. What are the key components of the quadratic formula?
3. How does initial velocity affect the trajectory of a falling object?
4. Can we apply this model for other values of initial velocity and height?
5. How does gravity (the constant -16 in this case) influence the motion of objects?

### Tip:
Always check the physical context of a problem, especially with time, to ensure negative or non-physical solutions are handled appropriately.

The height of a small rock falling from the top of a 124-ft-tall building with an initial downward velocity of −30 ft/sec is modeled by the equation h(t) = −16t² − 30t + 124, where t is the time in seconds. For which interval of time does the rock remain in the air?

Determine the interval during which a rock remains in the air when dropped from a 124-ft building with an initial velocity of -30 ft/sec. This problem involves solving a quadratic equation using the quadratic formula. Suitable for students in Grades 9-12.

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