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Answer :
To find out how long it takes the toy rocket to reach its maximum height and what that maximum height is, we can use the function provided:
[tex]\[ h(t) = -16t^2 + 72t + 4 \][/tex]
This is a quadratic function, where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 72 \)[/tex]
- [tex]\( c = 4 \)[/tex]
The maximum height of a quadratic function is reached at its vertex. For a parabola opening downwards (like this one, because [tex]\( a \)[/tex] is negative), the vertex's [tex]\( t \)[/tex]-coordinate can be found using the formula:
[tex]\[ t = \frac{-b}{2a} \][/tex]
Plugging in the values, we have:
[tex]\[ t = \frac{-72}{2 \times -16} = \frac{-72}{-32} = 2.25 \][/tex]
So, the rocket reaches its maximum height 2.25 seconds after launch.
To find the maximum height, substitute [tex]\( t = 2.25 \)[/tex] back into the height function:
[tex]\[ h(2.25) = -16(2.25)^2 + 72(2.25) + 4 \][/tex]
Calculating this will give:
[tex]\[ h(2.25) = -16(5.0625) + 162 + 4 = -81 + 162 + 4 = 85 \][/tex]
Therefore, the maximum height reached by the rocket is 85 feet.
So, in summary:
- The rocket reaches its maximum height 2.25 seconds after launch.
- The maximum height reached by the rocket is 85 feet.
[tex]\[ h(t) = -16t^2 + 72t + 4 \][/tex]
This is a quadratic function, where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 72 \)[/tex]
- [tex]\( c = 4 \)[/tex]
The maximum height of a quadratic function is reached at its vertex. For a parabola opening downwards (like this one, because [tex]\( a \)[/tex] is negative), the vertex's [tex]\( t \)[/tex]-coordinate can be found using the formula:
[tex]\[ t = \frac{-b}{2a} \][/tex]
Plugging in the values, we have:
[tex]\[ t = \frac{-72}{2 \times -16} = \frac{-72}{-32} = 2.25 \][/tex]
So, the rocket reaches its maximum height 2.25 seconds after launch.
To find the maximum height, substitute [tex]\( t = 2.25 \)[/tex] back into the height function:
[tex]\[ h(2.25) = -16(2.25)^2 + 72(2.25) + 4 \][/tex]
Calculating this will give:
[tex]\[ h(2.25) = -16(5.0625) + 162 + 4 = -81 + 162 + 4 = 85 \][/tex]
Therefore, the maximum height reached by the rocket is 85 feet.
So, in summary:
- The rocket reaches its maximum height 2.25 seconds after launch.
- The maximum height reached by the rocket is 85 feet.
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