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Answer :
Let's solve the problem step-by-step.
The height of the rocket above the ground is given by the equation:
[tex]\[ h(t) = -16t^2 + 32t + 48 \][/tex]
where [tex]\( t \)[/tex] is the time in seconds, and [tex]\( h(t) \)[/tex] is the height in feet.
### a. Find the maximum height of the rocket.
To find the maximum height, we need to determine the vertex of the parabola represented by the equation. The vertex form occurs when the derivative of the height equation is zero or using the vertex formula [tex]\( t = \frac{-b}{2a} \)[/tex].
Here, the equation is [tex]\( h(t) = -16t^2 + 32t + 48 \)[/tex].
- The coefficient [tex]\( a = -16 \)[/tex] and [tex]\( b = 32 \)[/tex].
Calculate the time at which the maximum height occurs:
[tex]\[ t = \frac{-b}{2a} = \frac{-32}{2 \times -16} = 1 \][/tex]
Now, plug this value back into the equation to find the maximum height:
[tex]\[ h(1) = -16(1)^2 + 32(1) + 48 \][/tex]
[tex]\[ h(1) = -16 + 32 + 48 \][/tex]
[tex]\[ h(1) = 64 \][/tex]
The maximum height of the rocket is 64 feet.
### b. Find the time it will take for the rocket to reach the ground.
To find when the rocket reaches the ground, solve for [tex]\( t \)[/tex] when [tex]\( h(t) = 0 \)[/tex].
The equation is:
[tex]\[ 0 = -16t^2 + 32t + 48 \][/tex]
Now solve this quadratic equation using the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 32^2 - 4 \times -16 \times 48 \][/tex]
[tex]\[ b^2 - 4ac = 1024 + 3072 = 4096 \][/tex]
Since the discriminant is positive, there are two real roots. Calculate the roots:
[tex]\[ t = \frac{-32 \pm \sqrt{4096}}{-32} \][/tex]
Solving for each root, we get:
[tex]\[ t_1 = \frac{-32 + 64}{-32} = -1 \][/tex] (not feasible as time cannot be negative)
[tex]\[ t_2 = \frac{-32 - 64}{-32} = 3 \][/tex]
The time it takes for the rocket to reach the ground is 3 seconds.
The height of the rocket above the ground is given by the equation:
[tex]\[ h(t) = -16t^2 + 32t + 48 \][/tex]
where [tex]\( t \)[/tex] is the time in seconds, and [tex]\( h(t) \)[/tex] is the height in feet.
### a. Find the maximum height of the rocket.
To find the maximum height, we need to determine the vertex of the parabola represented by the equation. The vertex form occurs when the derivative of the height equation is zero or using the vertex formula [tex]\( t = \frac{-b}{2a} \)[/tex].
Here, the equation is [tex]\( h(t) = -16t^2 + 32t + 48 \)[/tex].
- The coefficient [tex]\( a = -16 \)[/tex] and [tex]\( b = 32 \)[/tex].
Calculate the time at which the maximum height occurs:
[tex]\[ t = \frac{-b}{2a} = \frac{-32}{2 \times -16} = 1 \][/tex]
Now, plug this value back into the equation to find the maximum height:
[tex]\[ h(1) = -16(1)^2 + 32(1) + 48 \][/tex]
[tex]\[ h(1) = -16 + 32 + 48 \][/tex]
[tex]\[ h(1) = 64 \][/tex]
The maximum height of the rocket is 64 feet.
### b. Find the time it will take for the rocket to reach the ground.
To find when the rocket reaches the ground, solve for [tex]\( t \)[/tex] when [tex]\( h(t) = 0 \)[/tex].
The equation is:
[tex]\[ 0 = -16t^2 + 32t + 48 \][/tex]
Now solve this quadratic equation using the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 32^2 - 4 \times -16 \times 48 \][/tex]
[tex]\[ b^2 - 4ac = 1024 + 3072 = 4096 \][/tex]
Since the discriminant is positive, there are two real roots. Calculate the roots:
[tex]\[ t = \frac{-32 \pm \sqrt{4096}}{-32} \][/tex]
Solving for each root, we get:
[tex]\[ t_1 = \frac{-32 + 64}{-32} = -1 \][/tex] (not feasible as time cannot be negative)
[tex]\[ t_2 = \frac{-32 - 64}{-32} = 3 \][/tex]
The time it takes for the rocket to reach the ground is 3 seconds.
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