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Answer :
We start with the height function of the rocket given by
[tex]$$
s(t) = -16t^2 + 144t.
$$[/tex]
We want to determine the time(s) when the rocket is at 128 feet. This means we set
[tex]$$
-16t^2 + 144t = 128.
$$[/tex]
Step 1. Set the Equation to 0
Subtract 128 from both sides of the equation to obtain a quadratic equation:
[tex]$$
-16t^2 + 144t - 128 = 0.
$$[/tex]
Step 2. Simplify the Equation
Divide the entire equation by [tex]\(-16\)[/tex] to simplify it. We get
[tex]$$
\frac{-16t^2 + 144t - 128}{-16} = 0 \quad \Longrightarrow \quad t^2 - 9t + 8 = 0.
$$[/tex]
Step 3. Factor the Quadratic
Now, we factor the quadratic equation
[tex]$$
t^2 - 9t + 8 = 0.
$$[/tex]
We look for two numbers that multiply to [tex]\(8\)[/tex] and add to [tex]\(-9\)[/tex]. These numbers are [tex]\(-1\)[/tex] and [tex]\(-8\)[/tex]. Thus, the quadratic factors as
[tex]$$
(t - 1)(t - 8) = 0.
$$[/tex]
Step 4. Solve for [tex]\( t \)[/tex]
Set each factor equal to zero:
1. [tex]\( t - 1 = 0 \quad \Longrightarrow \quad t = 1, \)[/tex]
2. [tex]\( t - 8 = 0 \quad \Longrightarrow \quad t = 8. \)[/tex]
Conclusion
The rocket reaches 128 feet above the ground at
[tex]$$
t = 1 \text{ second} \quad \text{and} \quad t = 8 \text{ seconds}.
$$[/tex]
[tex]$$
s(t) = -16t^2 + 144t.
$$[/tex]
We want to determine the time(s) when the rocket is at 128 feet. This means we set
[tex]$$
-16t^2 + 144t = 128.
$$[/tex]
Step 1. Set the Equation to 0
Subtract 128 from both sides of the equation to obtain a quadratic equation:
[tex]$$
-16t^2 + 144t - 128 = 0.
$$[/tex]
Step 2. Simplify the Equation
Divide the entire equation by [tex]\(-16\)[/tex] to simplify it. We get
[tex]$$
\frac{-16t^2 + 144t - 128}{-16} = 0 \quad \Longrightarrow \quad t^2 - 9t + 8 = 0.
$$[/tex]
Step 3. Factor the Quadratic
Now, we factor the quadratic equation
[tex]$$
t^2 - 9t + 8 = 0.
$$[/tex]
We look for two numbers that multiply to [tex]\(8\)[/tex] and add to [tex]\(-9\)[/tex]. These numbers are [tex]\(-1\)[/tex] and [tex]\(-8\)[/tex]. Thus, the quadratic factors as
[tex]$$
(t - 1)(t - 8) = 0.
$$[/tex]
Step 4. Solve for [tex]\( t \)[/tex]
Set each factor equal to zero:
1. [tex]\( t - 1 = 0 \quad \Longrightarrow \quad t = 1, \)[/tex]
2. [tex]\( t - 8 = 0 \quad \Longrightarrow \quad t = 8. \)[/tex]
Conclusion
The rocket reaches 128 feet above the ground at
[tex]$$
t = 1 \text{ second} \quad \text{and} \quad t = 8 \text{ seconds}.
$$[/tex]
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