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Answer :
Let's solve the problem step by step:
The height of the toy rocket after [tex]\( t \)[/tex] seconds is given by the equation:
[tex]\[ h(t) = -16t^2 + 128t \][/tex]
a. How long will it take the rocket to hit its maximum height?
To find the time when the rocket reaches its maximum height, we need to find the vertex of the quadratic equation. The vertex formula for the time [tex]\( t \)[/tex] at maximum height is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
In the equation [tex]\( h(t) = -16t^2 + 128t \)[/tex]:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 128 \)[/tex]
Plug the values into the vertex formula:
[tex]\[ t = -\frac{128}{2 \times (-16)} \][/tex]
[tex]\[ t = \frac{128}{32} \][/tex]
[tex]\[ t = 4 \][/tex]
So, it takes 4 seconds for the rocket to reach its maximum height.
b. What is the maximum height?
To find the maximum height, substitute [tex]\( t = 4 \)[/tex] back into the height equation:
[tex]\[ h(4) = -16(4)^2 + 128(4) \][/tex]
[tex]\[ h(4) = -16 \times 16 + 512 \][/tex]
[tex]\[ h(4) = -256 + 512 \][/tex]
[tex]\[ h(4) = 256 \][/tex]
The maximum height the rocket reaches is 256 feet.
c. How long did it take for the rocket to reach the ground?
The rocket reaches the ground when the height [tex]\( h(t) = 0 \)[/tex]. We need to solve the equation:
[tex]\[ -16t^2 + 128t = 0 \][/tex]
Factor out [tex]\( t \)[/tex]:
[tex]\[ t(-16t + 128) = 0 \][/tex]
This gives us the solutions:
[tex]\[ t = 0 \quad \text{or} \quad -16t + 128 = 0 \][/tex]
Solving for [tex]\( t \)[/tex] when [tex]\( -16t + 128 = 0 \)[/tex]:
[tex]\[ -16t = -128 \][/tex]
[tex]\[ t = 8 \][/tex]
The rocket reaches the ground after 8 seconds.
In summary,
- The rocket takes 4 seconds to reach its maximum height.
- The maximum height is 256 feet.
- It takes 8 seconds for the rocket to reach the ground.
The height of the toy rocket after [tex]\( t \)[/tex] seconds is given by the equation:
[tex]\[ h(t) = -16t^2 + 128t \][/tex]
a. How long will it take the rocket to hit its maximum height?
To find the time when the rocket reaches its maximum height, we need to find the vertex of the quadratic equation. The vertex formula for the time [tex]\( t \)[/tex] at maximum height is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
In the equation [tex]\( h(t) = -16t^2 + 128t \)[/tex]:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 128 \)[/tex]
Plug the values into the vertex formula:
[tex]\[ t = -\frac{128}{2 \times (-16)} \][/tex]
[tex]\[ t = \frac{128}{32} \][/tex]
[tex]\[ t = 4 \][/tex]
So, it takes 4 seconds for the rocket to reach its maximum height.
b. What is the maximum height?
To find the maximum height, substitute [tex]\( t = 4 \)[/tex] back into the height equation:
[tex]\[ h(4) = -16(4)^2 + 128(4) \][/tex]
[tex]\[ h(4) = -16 \times 16 + 512 \][/tex]
[tex]\[ h(4) = -256 + 512 \][/tex]
[tex]\[ h(4) = 256 \][/tex]
The maximum height the rocket reaches is 256 feet.
c. How long did it take for the rocket to reach the ground?
The rocket reaches the ground when the height [tex]\( h(t) = 0 \)[/tex]. We need to solve the equation:
[tex]\[ -16t^2 + 128t = 0 \][/tex]
Factor out [tex]\( t \)[/tex]:
[tex]\[ t(-16t + 128) = 0 \][/tex]
This gives us the solutions:
[tex]\[ t = 0 \quad \text{or} \quad -16t + 128 = 0 \][/tex]
Solving for [tex]\( t \)[/tex] when [tex]\( -16t + 128 = 0 \)[/tex]:
[tex]\[ -16t = -128 \][/tex]
[tex]\[ t = 8 \][/tex]
The rocket reaches the ground after 8 seconds.
In summary,
- The rocket takes 4 seconds to reach its maximum height.
- The maximum height is 256 feet.
- It takes 8 seconds for the rocket to reach the ground.
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