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Answer :
To find out how long it takes for the rocket to reach its maximum height and what that height is, we can analyze the function representing the rocket's height: [tex]\( h(t) = -16t^2 + 144t + 5 \)[/tex]. This is a quadratic equation in the form [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 144 \)[/tex], and [tex]\( c = 5 \)[/tex].
### Step 1: Find the time when the rocket reaches its maximum height.
For a quadratic function in the form [tex]\( at^2 + bt + c \)[/tex], the maximum or minimum value (vertex) is obtained using the formula for the time [tex]\( t \)[/tex] at which this occurs:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plug in the values for [tex]\( b \)[/tex] and [tex]\( a \)[/tex]:
[tex]\[ t = -\frac{144}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{144}{-32} \][/tex]
[tex]\[ t = 4.5 \][/tex]
So, the rocket reaches its maximum height after 4.5 seconds.
### Step 2: Calculate the maximum height.
To find the maximum height, substitute [tex]\( t = 4.5 \)[/tex] back into the original height function [tex]\( h(t) \)[/tex]:
[tex]\[ h(4.5) = -16(4.5)^2 + 144(4.5) + 5 \][/tex]
Calculate this step-by-step:
1. [tex]\( (4.5)^2 = 20.25 \)[/tex]
2. [tex]\( -16 \times 20.25 = -324 \)[/tex]
3. [tex]\( 144 \times 4.5 = 648 \)[/tex]
4. Add these results along with the constant term:
[tex]\[ h(4.5) = -324 + 648 + 5 \][/tex]
[tex]\[ h(4.5) = 329 \][/tex]
The maximum height the rocket reaches is 329 feet.
In conclusion, the rocket reaches its maximum height at 4.5 seconds after launch, and the maximum height is 329 feet.
### Step 1: Find the time when the rocket reaches its maximum height.
For a quadratic function in the form [tex]\( at^2 + bt + c \)[/tex], the maximum or minimum value (vertex) is obtained using the formula for the time [tex]\( t \)[/tex] at which this occurs:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plug in the values for [tex]\( b \)[/tex] and [tex]\( a \)[/tex]:
[tex]\[ t = -\frac{144}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{144}{-32} \][/tex]
[tex]\[ t = 4.5 \][/tex]
So, the rocket reaches its maximum height after 4.5 seconds.
### Step 2: Calculate the maximum height.
To find the maximum height, substitute [tex]\( t = 4.5 \)[/tex] back into the original height function [tex]\( h(t) \)[/tex]:
[tex]\[ h(4.5) = -16(4.5)^2 + 144(4.5) + 5 \][/tex]
Calculate this step-by-step:
1. [tex]\( (4.5)^2 = 20.25 \)[/tex]
2. [tex]\( -16 \times 20.25 = -324 \)[/tex]
3. [tex]\( 144 \times 4.5 = 648 \)[/tex]
4. Add these results along with the constant term:
[tex]\[ h(4.5) = -324 + 648 + 5 \][/tex]
[tex]\[ h(4.5) = 329 \][/tex]
The maximum height the rocket reaches is 329 feet.
In conclusion, the rocket reaches its maximum height at 4.5 seconds after launch, and the maximum height is 329 feet.
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