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Answer :
To find out how long it will take for the rocket to hit the ground, we need to determine when its height [tex]\( h \)[/tex] will be zero. The height of the rocket is described by the quadratic equation:
[tex]\[ h = -2t^2 + 7t + 4 \][/tex]
Where [tex]\( t \)[/tex] is the time in seconds. We set [tex]\( h = 0 \)[/tex] and solve for [tex]\( t \)[/tex]:
[tex]\[ 0 = -2t^2 + 7t + 4 \][/tex]
This is a quadratic equation in the standard form [tex]\( at^2 + bt + c = 0 \)[/tex], where [tex]\( a = -2 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = 4 \)[/tex].
To solve this equation, we can use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step-by-step solution:
1. Calculate the discriminant:
The discriminant is given by [tex]\( b^2 - 4ac \)[/tex].
[tex]\[
b^2 = 7^2 = 49
\][/tex]
[tex]\[
4ac = 4 \times (-2) \times 4 = -32
\][/tex]
[tex]\[
\text{Discriminant} = 49 - (-32) = 49 + 32 = 81
\][/tex]
2. Solve for [tex]\( t \)[/tex] using the quadratic formula:
Since we have a positive discriminant, there will be two real solutions.
[tex]\[
t = \frac{-7 \pm \sqrt{81}}{2 \times -2}
\][/tex]
Calculate [tex]\( \sqrt{81} \)[/tex]:
[tex]\[
\sqrt{81} = 9
\][/tex]
Possible solutions for [tex]\( t \)[/tex]:
[tex]\[
t_1 = \frac{-7 + 9}{-4} = \frac{2}{-4} = -0.5
\][/tex]
[tex]\[
t_2 = \frac{-7 - 9}{-4} = \frac{-16}{-4} = 4.0
\][/tex]
3. Choose the valid time:
As time cannot be negative, we discard [tex]\( t = -0.5 \)[/tex].
Therefore, the rocket will hit the ground at [tex]\( t = 4.0 \)[/tex] seconds.
So, it will take 4.0 seconds for the rocket to hit the ground.
[tex]\[ h = -2t^2 + 7t + 4 \][/tex]
Where [tex]\( t \)[/tex] is the time in seconds. We set [tex]\( h = 0 \)[/tex] and solve for [tex]\( t \)[/tex]:
[tex]\[ 0 = -2t^2 + 7t + 4 \][/tex]
This is a quadratic equation in the standard form [tex]\( at^2 + bt + c = 0 \)[/tex], where [tex]\( a = -2 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = 4 \)[/tex].
To solve this equation, we can use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step-by-step solution:
1. Calculate the discriminant:
The discriminant is given by [tex]\( b^2 - 4ac \)[/tex].
[tex]\[
b^2 = 7^2 = 49
\][/tex]
[tex]\[
4ac = 4 \times (-2) \times 4 = -32
\][/tex]
[tex]\[
\text{Discriminant} = 49 - (-32) = 49 + 32 = 81
\][/tex]
2. Solve for [tex]\( t \)[/tex] using the quadratic formula:
Since we have a positive discriminant, there will be two real solutions.
[tex]\[
t = \frac{-7 \pm \sqrt{81}}{2 \times -2}
\][/tex]
Calculate [tex]\( \sqrt{81} \)[/tex]:
[tex]\[
\sqrt{81} = 9
\][/tex]
Possible solutions for [tex]\( t \)[/tex]:
[tex]\[
t_1 = \frac{-7 + 9}{-4} = \frac{2}{-4} = -0.5
\][/tex]
[tex]\[
t_2 = \frac{-7 - 9}{-4} = \frac{-16}{-4} = 4.0
\][/tex]
3. Choose the valid time:
As time cannot be negative, we discard [tex]\( t = -0.5 \)[/tex].
Therefore, the rocket will hit the ground at [tex]\( t = 4.0 \)[/tex] seconds.
So, it will take 4.0 seconds for the rocket to hit the ground.
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