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We are given that a firework is launched with an initial velocity of \( v_0 = 128 \, \text{ft/s} \) from ground level. The acceleration due to gravity is \( a = -16 \, \text{ft/s}^2 \), and we are tasked with finding the maximum height the firework reaches.

The height \( h(t) \) as a function of time is given by the formula:
\[
h(t) = at^2 + v_0 t + h_0
\]
Where:
- \( a = -16 \, \text{ft/s}^2 \) (gravity)
- \( v_0 = 128 \, \text{ft/s} \) (initial velocity)
- \( h_0 = 0 \, \text{ft} \) (since the firework starts from ground level)

### Step 1: Find the time when the firework reaches its maximum height
At maximum height, the velocity is zero. Velocity \( v(t) \) is the derivative of the height function:
\[
v(t) = \frac{dh}{dt} = 2at + v_0
\]
Set \( v(t) = 0 \) to find the time when the firework reaches the maximum height:
\[
0 = 2(-16)t + 128
\]
\[
-32t + 128 = 0
\]
\[
t = \frac{128}{32} = 4 \, \text{seconds}
\]

### Step 2: Find the maximum height
Now substitute \( t = 4 \) seconds into the height equation to find the maximum height:
\[
h(4) = -16(4)^2 + 128(4) + 0
\]
\[
h(4) = -16(16) + 128(4)
\]
\[
h(4) = -256 + 512
\]
\[
h(4) = 256 \, \text{ft}
\]

Thus, the maximum height reached by the firework is **256 feet**, which corresponds to one of the provided answer choices.

Would you like further details on this or any follow-up questions?

Here are 5 related questions to expand on the topic:
1. How would the maximum height change if the initial velocity was doubled?
2. What would the maximum height be if the firework was launched from an elevated platform 50 ft above ground?
3. How would you find the time it takes for the firework to hit the ground?
4. How does air resistance affect the height and time of flight of a firework?
5. What are the units for acceleration, and how do they relate to velocity and height?

**Tip:** When dealing with free-fall motion, remember that the acceleration due to gravity is always negative since it pulls objects downward.

A firework is launched into the air from ground level with an initial velocity of 128 ft/s. If acceleration due to gravity is -16 ft/s², what is the maximum height reached by the firework?

Learn how to calculate the maximum height of a firework launched with an initial velocity of 128 ft/s under the influence of gravity (-16 ft/s²). The detailed solution involves using kinematic equations and derivatives to find the height of 256 feet, suitable for high school students.

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