The function \( h(t) = -16t^2 + 96t \) models the height, in feet, of a firework \( t \) seconds after it has been launched. To analyze this function, we can identify several key characteristics:
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Vertex of the Parabola: The function is a quadratic equation in standard form \( h(t) = at^2 + bt + c \), where \( a = -16 \), \( b = 96 \), and \( c = 0 \). The vertex of the parabola can be found using the formula \( t = -\frac{b}{2a} \).
\[ t = -\frac{96}{2 \times -16} = -\frac{96}{-32} = 3 \text{ seconds} \]
To find the maximum height, we substitute \( t = 3 \) back into the height function:
\[ h(3) = -16(3^2) + 96(3) \] \[ h(3) = -16(9) + 288 \] \[ h(3) = -144 + 288 = 144 \text{ feet} \]
Thus, the maximum height of the firework is 144 feet, reached at 3 seconds after launch.
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Finding the Times When the Firework Hits the Ground: To determine when the firework hits the ground, we set \( h(t) = 0 \) and solve for \( t \):
\[ -16t^2 + 96t = 0 \]
Factoring out common terms:
\[ -16t(t - 6) = 0 \]
Setting each factor equal to zero gives:
\[ t = 0 \quad \text{or} \quad t - 6 = 0 \implies t = 6 \]
Therefore, the firework is launched at \( t = 0 \) seconds and hits the ground again at \( t = 6 \) seconds.
In summary:
- The firework reaches a maximum height of 144 feet at 3 seconds after launch.
- It hits the ground again at 6 seconds after launch.