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Answer :
To find the time it takes for the toy rocket to reach its maximum height and to determine what that maximum height is, we use the function given for the height of the rocket:
[tex]\[ h(t) = -16t^2 + 88t + 6 \][/tex]
This is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 88 \)[/tex], and [tex]\( c = 6 \)[/tex].
### Step 1: Find the Time to Reach Maximum Height
The maximum height of a quadratic function occurs at its vertex. For a quadratic equation [tex]\( ax^2 + bx + c \)[/tex], the formula to find the time [tex]\( t \)[/tex] at which the vertex occurs (and thus the maximum height, since the parabola opens downwards) is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the given values:
[tex]\( a = -16 \)[/tex]
[tex]\( b = 88 \)[/tex]
[tex]\[ t = -\frac{88}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{88}{-32} \][/tex]
[tex]\[ t = \frac{88}{32} \][/tex]
[tex]\[ t = 2.75 \][/tex]
So, it takes 2.75 seconds for the rocket to reach its maximum height.
### Step 2: Calculate the Maximum Height
Now, substitute [tex]\( t = 2.75 \)[/tex] into the height function to find the maximum height:
[tex]\[ h(2.75) = -16(2.75)^2 + 88(2.75) + 6 \][/tex]
Calculate each term step by step:
1. [tex]\( (2.75)^2 = 7.5625 \)[/tex]
2. [tex]\( -16 \times 7.5625 = -121 \)[/tex]
3. [tex]\( 88 \times 2.75 = 242 \)[/tex]
Finally, substitute these into the equation:
[tex]\[ h(2.75) = -121 + 242 + 6 \][/tex]
[tex]\[ h(2.75) = 127 \][/tex]
Therefore, the maximum height reached by the rocket is 127 feet.
### Conclusion
The rocket reaches its maximum height 2.75 seconds after launch, and the maximum height is 127 feet.
[tex]\[ h(t) = -16t^2 + 88t + 6 \][/tex]
This is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 88 \)[/tex], and [tex]\( c = 6 \)[/tex].
### Step 1: Find the Time to Reach Maximum Height
The maximum height of a quadratic function occurs at its vertex. For a quadratic equation [tex]\( ax^2 + bx + c \)[/tex], the formula to find the time [tex]\( t \)[/tex] at which the vertex occurs (and thus the maximum height, since the parabola opens downwards) is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging in the given values:
[tex]\( a = -16 \)[/tex]
[tex]\( b = 88 \)[/tex]
[tex]\[ t = -\frac{88}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{88}{-32} \][/tex]
[tex]\[ t = \frac{88}{32} \][/tex]
[tex]\[ t = 2.75 \][/tex]
So, it takes 2.75 seconds for the rocket to reach its maximum height.
### Step 2: Calculate the Maximum Height
Now, substitute [tex]\( t = 2.75 \)[/tex] into the height function to find the maximum height:
[tex]\[ h(2.75) = -16(2.75)^2 + 88(2.75) + 6 \][/tex]
Calculate each term step by step:
1. [tex]\( (2.75)^2 = 7.5625 \)[/tex]
2. [tex]\( -16 \times 7.5625 = -121 \)[/tex]
3. [tex]\( 88 \times 2.75 = 242 \)[/tex]
Finally, substitute these into the equation:
[tex]\[ h(2.75) = -121 + 242 + 6 \][/tex]
[tex]\[ h(2.75) = 127 \][/tex]
Therefore, the maximum height reached by the rocket is 127 feet.
### Conclusion
The rocket reaches its maximum height 2.75 seconds after launch, and the maximum height is 127 feet.
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