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From the image you uploaded, I can see the problem involves using the kinematic equation for the motion of a projectile:

\[ s = -16t^2 + v_0t + s_0 \]

where:
- \( s \) represents the height of an object in feet.
- \( v_0 \) is the initial velocity of the object in feet per second.
- \( s_0 \) is the initial height of the object (in feet).
- \( t \) is the time in seconds.

Given in the problem:
- \( v_0 = 96 \) feet per second (initial velocity)
- \( s_0 = 0 \) (launched from ground level)

### Part (a): When the projectile is back at ground level
To find when the projectile returns to ground level, set \( s = 0 \) and solve for \( t \):
\[ 0 = -16t^2 + 96t \]
\[ -16t^2 + 96t = 0 \]
Factoring out \( t \):
\[ t(-16t + 96) = 0 \]
Setting each factor equal to zero gives:
\[ t = 0 \quad \text{(at launch)} \]
\[ -16t + 96 = 0 \]
\[ t = 6 \] seconds

### Part (b): When the height exceeds 128 feet
Solve the inequality:
\[ -16t^2 + 96t > 128 \]
\[ -16t^2 + 96t - 128 > 0 \]
\[ 16t^2 - 96t + 128 < 0 \]
Factoring or using the quadratic formula:
\[ (4t - 8)^2 = 0 \]
\[ 4t - 8 = 0 \]
\[ t = 2 \] seconds (when it reaches 128 feet exactly)

For the interval when the height exceeds 128 feet, since this is a quadratic that opens downwards (parabola peaking at its vertex), and it equals 128 feet at \( t = 2 \) seconds, it will exceed 128 feet between the time it ascends past and descends back through this height:
\[ 0 < t < 4 \]

Therefore:
1. The projectile is back on the ground at \( t = 6 \) seconds.
2. The projectile's height exceeds 128 feet during \( (0, 4) \) seconds.

Do you need more detailed steps on any part of this problem or have any other questions? Here are some related questions you might consider:

1. What is the maximum height reached by the projectile?
2. At what time does the projectile reach its maximum height?
3. How would the time of flight change if the initial velocity was increased or decreased?
4. What would the height-time graph of this projectile look like?
5. How do factors like air resistance impact the actual flight path of a projectile compared to this idealized scenario?

**Tip**: To check if a quadratic function is increasing or decreasing at a particular point, consider calculating the first derivative and evaluating it at that point. This tells you the slope (rate of change) of the function at that point.

A projectile is fired straight upward from ground level (s_0 = 0) with an initial velocity of 96 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 128 feet?

Solve a projectile motion problem involving a projectile fired upward at 96 feet per second. Learn how to calculate when the projectile will return to the ground and when its height will exceed 128 feet using quadratic equations.

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