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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, you can use the given quadratic function for the height of the rocket:
[tex]\[ h(t) = -16t^2 + 32t + 9, \][/tex]
where [tex]\( h(t) \)[/tex] is the height in feet and [tex]\( t \)[/tex] is the time in seconds after launch.
### Step 1: Determine the Time to Reach Maximum Height
A quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex] reaches its maximum (or minimum) at the vertex. For a parabola opening downwards (like this one, since the coefficient of [tex]\( t^2 \)[/tex] is negative), the maximum value occurs at:
[tex]\[ t = -\frac{b}{2a}. \][/tex]
Here, [tex]\( a = -16 \)[/tex] and [tex]\( b = 32 \)[/tex].
Plug these values into the formula:
[tex]\[ t = -\frac{32}{2 \times -16} = 1 \, \text{second}. \][/tex]
So, the rocket reaches its maximum height 1 second after launch.
### Step 2: Calculate the Maximum Height
Now, substitute [tex]\( t = 1 \)[/tex] back into the original height function to find the maximum height:
[tex]\[ h(1) = -16(1)^2 + 32(1) + 9. \][/tex]
Perform the calculations step by step:
[tex]\[ h(1) = -16(1) + 32 + 9, \][/tex]
[tex]\[ h(1) = -16 + 32 + 9, \][/tex]
[tex]\[ h(1) = 25 \, \text{feet}. \][/tex]
Therefore, the maximum height of the rocket is 25 feet.
In summary, the rocket reaches its maximum height 1 second after launch, and the maximum height is 25 feet.
[tex]\[ h(t) = -16t^2 + 32t + 9, \][/tex]
where [tex]\( h(t) \)[/tex] is the height in feet and [tex]\( t \)[/tex] is the time in seconds after launch.
### Step 1: Determine the Time to Reach Maximum Height
A quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex] reaches its maximum (or minimum) at the vertex. For a parabola opening downwards (like this one, since the coefficient of [tex]\( t^2 \)[/tex] is negative), the maximum value occurs at:
[tex]\[ t = -\frac{b}{2a}. \][/tex]
Here, [tex]\( a = -16 \)[/tex] and [tex]\( b = 32 \)[/tex].
Plug these values into the formula:
[tex]\[ t = -\frac{32}{2 \times -16} = 1 \, \text{second}. \][/tex]
So, the rocket reaches its maximum height 1 second after launch.
### Step 2: Calculate the Maximum Height
Now, substitute [tex]\( t = 1 \)[/tex] back into the original height function to find the maximum height:
[tex]\[ h(1) = -16(1)^2 + 32(1) + 9. \][/tex]
Perform the calculations step by step:
[tex]\[ h(1) = -16(1) + 32 + 9, \][/tex]
[tex]\[ h(1) = -16 + 32 + 9, \][/tex]
[tex]\[ h(1) = 25 \, \text{feet}. \][/tex]
Therefore, the maximum height of the rocket is 25 feet.
In summary, the rocket reaches its maximum height 1 second after launch, and the maximum height is 25 feet.
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