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Answer :
To find out when the rocket will hit the ground, you need to determine when its height [tex]\( h(t) \)[/tex] becomes zero. The equation given for the rocket's height is:
[tex]\[ h(t) = -16t^2 + 96t \][/tex]
You want to find the value of [tex]\( t \)[/tex] for which the height is zero:
[tex]\[ -16t^2 + 96t = 0 \][/tex]
This is a quadratic equation that can be solved by factoring. Let's factor the equation step-by-step:
1. Factor out the greatest common factor, which is [tex]\( 16t \)[/tex]:
[tex]\[ 16t(-t + 6) = 0 \][/tex]
2. Set each factor equal to zero:
[tex]\[ 16t = 0 \quad \text{or} \quad -t + 6 = 0 \][/tex]
3. Solve each equation:
- For [tex]\( 16t = 0 \)[/tex]:
[tex]\( t = 0 \)[/tex]
- For [tex]\( -t + 6 = 0 \)[/tex]:
[tex]\[ -t + 6 = 0 \][/tex]
Add [tex]\( t \)[/tex] to both sides:
[tex]\( 6 = t \)[/tex]
So, [tex]\( t = 6 \)[/tex]
The solutions for [tex]\( t \)[/tex] are [tex]\( t = 0 \)[/tex] and [tex]\( t = 6 \)[/tex].
In the context of this problem:
- [tex]\( t = 0 \)[/tex] corresponds to the time when the rocket was initially launched.
- [tex]\( t = 6 \)[/tex] is the time when the rocket hits the ground.
Therefore, the rocket will hit the ground after 6 seconds.
[tex]\[ h(t) = -16t^2 + 96t \][/tex]
You want to find the value of [tex]\( t \)[/tex] for which the height is zero:
[tex]\[ -16t^2 + 96t = 0 \][/tex]
This is a quadratic equation that can be solved by factoring. Let's factor the equation step-by-step:
1. Factor out the greatest common factor, which is [tex]\( 16t \)[/tex]:
[tex]\[ 16t(-t + 6) = 0 \][/tex]
2. Set each factor equal to zero:
[tex]\[ 16t = 0 \quad \text{or} \quad -t + 6 = 0 \][/tex]
3. Solve each equation:
- For [tex]\( 16t = 0 \)[/tex]:
[tex]\( t = 0 \)[/tex]
- For [tex]\( -t + 6 = 0 \)[/tex]:
[tex]\[ -t + 6 = 0 \][/tex]
Add [tex]\( t \)[/tex] to both sides:
[tex]\( 6 = t \)[/tex]
So, [tex]\( t = 6 \)[/tex]
The solutions for [tex]\( t \)[/tex] are [tex]\( t = 0 \)[/tex] and [tex]\( t = 6 \)[/tex].
In the context of this problem:
- [tex]\( t = 0 \)[/tex] corresponds to the time when the rocket was initially launched.
- [tex]\( t = 6 \)[/tex] is the time when the rocket hits the ground.
Therefore, the rocket will hit the ground after 6 seconds.
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