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Answer :
To find out how long it will take for the toy rocket to return to the ground, we start with the height equation given by:
[tex]\[ h(t) = -16t^2 + 148t \][/tex]
The rocket returns to the ground when the height [tex]\( h(t) \)[/tex] is 0. Therefore, we need to solve the equation:
[tex]\[ -16t^2 + 148t = 0 \][/tex]
First, factor out the common factor of [tex]\( t \)[/tex]:
[tex]\[ t(-16t + 148) = 0 \][/tex]
This factored equation gives us two possible solutions:
1. [tex]\( t = 0 \)[/tex]
2. [tex]\( -16t + 148 = 0 \)[/tex]
The solution [tex]\( t = 0 \)[/tex] represents the initial time when the rocket was launched from the ground.
Next, we solve the equation [tex]\( -16t + 148 = 0 \)[/tex] for [tex]\( t \)[/tex]:
[tex]\[ -16t + 148 = 0 \][/tex]
Subtract 148 from both sides to isolate terms involving [tex]\( t \)[/tex]:
[tex]\[ -16t = -148 \][/tex]
Divide both sides by -16 to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{148}{16} \][/tex]
Simplifying:
[tex]\[ t = 9.25 \][/tex]
Therefore, the rocket will take 9.25 seconds to return to the ground after being launched.
[tex]\[ h(t) = -16t^2 + 148t \][/tex]
The rocket returns to the ground when the height [tex]\( h(t) \)[/tex] is 0. Therefore, we need to solve the equation:
[tex]\[ -16t^2 + 148t = 0 \][/tex]
First, factor out the common factor of [tex]\( t \)[/tex]:
[tex]\[ t(-16t + 148) = 0 \][/tex]
This factored equation gives us two possible solutions:
1. [tex]\( t = 0 \)[/tex]
2. [tex]\( -16t + 148 = 0 \)[/tex]
The solution [tex]\( t = 0 \)[/tex] represents the initial time when the rocket was launched from the ground.
Next, we solve the equation [tex]\( -16t + 148 = 0 \)[/tex] for [tex]\( t \)[/tex]:
[tex]\[ -16t + 148 = 0 \][/tex]
Subtract 148 from both sides to isolate terms involving [tex]\( t \)[/tex]:
[tex]\[ -16t = -148 \][/tex]
Divide both sides by -16 to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{148}{16} \][/tex]
Simplifying:
[tex]\[ t = 9.25 \][/tex]
Therefore, the rocket will take 9.25 seconds to return to the ground after being launched.
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