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Answer :
To determine how long it will take for the toy rocket to reach the ground, we need to solve the equation for the rocket's height [tex]\( h(t) \)[/tex] when it is equal to zero. The equation given for the height of the rocket at any time [tex]\( t \)[/tex] is:
[tex]\[ h(t) = -16t^2 + 128t + 12 \][/tex]
The rocket reaches the ground when its height [tex]\( h(t) \)[/tex] is 0. Therefore, we need to solve the equation:
[tex]\[ -16t^2 + 128t + 12 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( at^2 + bt + c = 0 \)[/tex], where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 128 \)[/tex]
- [tex]\( c = 12 \)[/tex]
To solve for [tex]\( t \)[/tex], we'll use the quadratic formula:
[tex]\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Let's calculate the discriminant [tex]\( \Delta = b^2 - 4ac \)[/tex]:
- [tex]\( b^2 = 128^2 = 16384 \)[/tex]
- [tex]\( 4ac = 4 \times (-16) \times 12 = -768 \)[/tex]
Therefore, the discriminant is:
[tex]\[
b^2 - 4ac = 16384 + 768 = 17152
\][/tex]
Now, substitute the values into the quadratic formula:
[tex]\[
t = \frac{-128 \pm \sqrt{17152}}{-32}
\][/tex]
Calculate the two possible values for [tex]\( t \)[/tex]:
- The positive root:
[tex]\[
t = \frac{-128 + \sqrt{17152}}{-32}
\][/tex]
The answer simplifies to:
[tex]\[
t \approx 4 + \frac{\sqrt{67}}{2}
\][/tex]
This is the positive time when the rocket will reach the ground, as time cannot be negative. Therefore, this is the time it takes for the rocket to reach the ground.
[tex]\[ h(t) = -16t^2 + 128t + 12 \][/tex]
The rocket reaches the ground when its height [tex]\( h(t) \)[/tex] is 0. Therefore, we need to solve the equation:
[tex]\[ -16t^2 + 128t + 12 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( at^2 + bt + c = 0 \)[/tex], where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 128 \)[/tex]
- [tex]\( c = 12 \)[/tex]
To solve for [tex]\( t \)[/tex], we'll use the quadratic formula:
[tex]\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Let's calculate the discriminant [tex]\( \Delta = b^2 - 4ac \)[/tex]:
- [tex]\( b^2 = 128^2 = 16384 \)[/tex]
- [tex]\( 4ac = 4 \times (-16) \times 12 = -768 \)[/tex]
Therefore, the discriminant is:
[tex]\[
b^2 - 4ac = 16384 + 768 = 17152
\][/tex]
Now, substitute the values into the quadratic formula:
[tex]\[
t = \frac{-128 \pm \sqrt{17152}}{-32}
\][/tex]
Calculate the two possible values for [tex]\( t \)[/tex]:
- The positive root:
[tex]\[
t = \frac{-128 + \sqrt{17152}}{-32}
\][/tex]
The answer simplifies to:
[tex]\[
t \approx 4 + \frac{\sqrt{67}}{2}
\][/tex]
This is the positive time when the rocket will reach the ground, as time cannot be negative. Therefore, this is the time it takes for the rocket to reach the ground.
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