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Answer :
Certainly! To find out when the rocket reaches its maximum height and what that height is, we start by analyzing the quadratic function given for the height of the rocket:
[tex]\[ h(t) = -16t^2 + 96t + 4 \][/tex]
This is a quadratic equation of the form [tex]\( at^2 + bt + c \)[/tex].
### Step 1: Find the time it takes to reach maximum height.
The rocket reaches its maximum height at the vertex of the parabola described by the quadratic equation. The vertex form for the time [tex]\( t \)[/tex] at which maximum height is reached is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
For the given equation [tex]\( h(t) = -16t^2 + 96t + 4 \)[/tex], the coefficients are:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 96 \)[/tex]
Plug these values into the formula:
[tex]\[ t = -\frac{96}{2(-16)} \][/tex]
[tex]\[ t = -\frac{96}{-32} \][/tex]
[tex]\[ t = 3 \][/tex]
So, the rocket reaches its maximum height 3 seconds after launch.
### Step 2: Calculate the maximum height.
Now, substitute [tex]\( t = 3 \)[/tex] back into the height function to find the maximum height [tex]\( h(3) \)[/tex]:
[tex]\[ h(3) = -16(3)^2 + 96(3) + 4 \][/tex]
[tex]\[ h(3) = -16 \times 9 + 96 \times 3 + 4 \][/tex]
[tex]\[ h(3) = -144 + 288 + 4 \][/tex]
[tex]\[ h(3) = 148 \][/tex]
Thus, the maximum height reached by the rocket is 148 feet.
Final Answer:
The rocket reaches its maximum height 3 seconds after launch, and the maximum height is 148 feet.
[tex]\[ h(t) = -16t^2 + 96t + 4 \][/tex]
This is a quadratic equation of the form [tex]\( at^2 + bt + c \)[/tex].
### Step 1: Find the time it takes to reach maximum height.
The rocket reaches its maximum height at the vertex of the parabola described by the quadratic equation. The vertex form for the time [tex]\( t \)[/tex] at which maximum height is reached is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
For the given equation [tex]\( h(t) = -16t^2 + 96t + 4 \)[/tex], the coefficients are:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 96 \)[/tex]
Plug these values into the formula:
[tex]\[ t = -\frac{96}{2(-16)} \][/tex]
[tex]\[ t = -\frac{96}{-32} \][/tex]
[tex]\[ t = 3 \][/tex]
So, the rocket reaches its maximum height 3 seconds after launch.
### Step 2: Calculate the maximum height.
Now, substitute [tex]\( t = 3 \)[/tex] back into the height function to find the maximum height [tex]\( h(3) \)[/tex]:
[tex]\[ h(3) = -16(3)^2 + 96(3) + 4 \][/tex]
[tex]\[ h(3) = -16 \times 9 + 96 \times 3 + 4 \][/tex]
[tex]\[ h(3) = -144 + 288 + 4 \][/tex]
[tex]\[ h(3) = 148 \][/tex]
Thus, the maximum height reached by the rocket is 148 feet.
Final Answer:
The rocket reaches its maximum height 3 seconds after launch, and the maximum height is 148 feet.
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