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Answer :
To find out how long it will take the toy rocket to reach its maximum height, and what that maximum height is, we use the given quadratic height function [tex]\( h(t) = -16t^2 + 56t + 9 \)[/tex].
### Finding the Time for Maximum Height:
The height function is a quadratic equation in the form [tex]\( at^2 + bt + c \)[/tex], which is a parabola. The graph of a parabola opens downwards when the coefficient [tex]\( a \)[/tex] (which is -16 in this case) is negative, which means it has a maximum point.
To find the time [tex]\( t \)[/tex] at which the rocket reaches its maximum height, we use the formula for the vertex of a parabola:
[tex]\[ t = -\frac{b}{2a} \][/tex]
For this equation:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 56 \)[/tex]
Plug these values into the formula:
[tex]\[ t = -\frac{56}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{56}{-32} \][/tex]
[tex]\[ t = \frac{56}{32} \][/tex]
[tex]\[ t = 1.75 \][/tex]
So, the rocket reaches its maximum height 1.75 seconds after launch.
### Finding the Maximum Height:
To find the maximum height of the rocket, substitute [tex]\( t = 1.75 \)[/tex] back into the height function:
[tex]\[ h(1.75) = -16(1.75)^2 + 56(1.75) + 9 \][/tex]
First, calculate [tex]\( (1.75)^2 \)[/tex]:
[tex]\[ (1.75)^2 = 3.0625 \][/tex]
Now substitute:
[tex]\[ h(1.75) = -16 \times 3.0625 + 56 \times 1.75 + 9 \][/tex]
Calculate each term:
[tex]\[ = -49 + 98 + 9 \][/tex]
[tex]\[ = 58 \][/tex]
Thus, the maximum height of the rocket is 58 feet.
In summary:
- The rocket reaches its maximum height 1.75 seconds after launch.
- The maximum height of the rocket is 58 feet.
### Finding the Time for Maximum Height:
The height function is a quadratic equation in the form [tex]\( at^2 + bt + c \)[/tex], which is a parabola. The graph of a parabola opens downwards when the coefficient [tex]\( a \)[/tex] (which is -16 in this case) is negative, which means it has a maximum point.
To find the time [tex]\( t \)[/tex] at which the rocket reaches its maximum height, we use the formula for the vertex of a parabola:
[tex]\[ t = -\frac{b}{2a} \][/tex]
For this equation:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 56 \)[/tex]
Plug these values into the formula:
[tex]\[ t = -\frac{56}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{56}{-32} \][/tex]
[tex]\[ t = \frac{56}{32} \][/tex]
[tex]\[ t = 1.75 \][/tex]
So, the rocket reaches its maximum height 1.75 seconds after launch.
### Finding the Maximum Height:
To find the maximum height of the rocket, substitute [tex]\( t = 1.75 \)[/tex] back into the height function:
[tex]\[ h(1.75) = -16(1.75)^2 + 56(1.75) + 9 \][/tex]
First, calculate [tex]\( (1.75)^2 \)[/tex]:
[tex]\[ (1.75)^2 = 3.0625 \][/tex]
Now substitute:
[tex]\[ h(1.75) = -16 \times 3.0625 + 56 \times 1.75 + 9 \][/tex]
Calculate each term:
[tex]\[ = -49 + 98 + 9 \][/tex]
[tex]\[ = 58 \][/tex]
Thus, the maximum height of the rocket is 58 feet.
In summary:
- The rocket reaches its maximum height 1.75 seconds after launch.
- The maximum height of the rocket is 58 feet.
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