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To solve the problem involving the height of a rocket given by the function [tex]\( h(t) = -16t^2 + 144t + 576 \)[/tex], we need to find the maximum height, the time it takes to reach that maximum height, and the time at which the rocket reaches the ground.
### Finding the Maximum Height
The height function is a quadratic equation of the form [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\(a = -16\)[/tex], [tex]\(b = 144\)[/tex], and [tex]\(c = 576\)[/tex]. This parabola opens downwards (since [tex]\(a\)[/tex] is negative), and the maximum height occurs at the vertex of the parabola.
The vertex of a parabola given by [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ t_{\text{max}} = -\frac{b}{2a} \][/tex]
Substituting the coefficients:
[tex]\[ t_{\text{max}} = -\frac{144}{2 \times -16} \][/tex]
[tex]\[ t_{\text{max}} = -\frac{144}{-32} \][/tex]
[tex]\[ t_{\text{max}} = 4.5 \][/tex]
To find the maximum height, we substitute [tex]\( t_{\text{max}} \)[/tex] back into the height function:
[tex]\[ h(4.5) = -16(4.5)^2 + 144(4.5) + 576 \][/tex]
[tex]\[ h(4.5) = -16(20.25) + 144(4.5) + 576 \][/tex]
[tex]\[ h(4.5) = -324 + 648 + 576 \][/tex]
[tex]\[ h(4.5) = 900 \][/tex]
Thus, the maximum height of the rocket is 900 feet, and it occurs at [tex]\( t = 4.5 \)[/tex] seconds.
### Finding the Time When the Rocket Reaches the Ground
The rocket reaches the ground when the height is zero:
[tex]\[ -16t^2 + 144t + 576 = 0 \][/tex]
This is a quadratic equation, which we can solve using the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the coefficients:
[tex]\[ a = -16 \][/tex]
[tex]\[ b = 144 \][/tex]
[tex]\[ c = 576 \][/tex]
[tex]\[ t = \frac{-144 \pm \sqrt{144^2 - 4(-16)(576)}}{2(-16)} \][/tex]
[tex]\[ t = \frac{-144 \pm \sqrt{20736 + 36864}}{-32} \][/tex]
[tex]\[ t = \frac{-144 \pm \sqrt{57600}}{-32} \][/tex]
[tex]\[ t = \frac{-144 \pm 240}{-32} \][/tex]
This gives us two solutions:
[tex]\[ t = \frac{96}{-32} = -3 \quad \text{(not physically meaningful, since time cannot be negative)} \][/tex]
[tex]\[ t = \frac{-384}{-32} = 12 \][/tex]
Thus, the rocket reaches the ground at [tex]\( t = 12 \)[/tex] seconds.
### Summary
- The rocket's maximum height is 900 feet.
- It takes 4.5 seconds for the rocket to reach its maximum height.
- The rocket reaches the ground at 12 seconds after it is fired.
These are the detailed steps needed to solve the problem correctly.
### Finding the Maximum Height
The height function is a quadratic equation of the form [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\(a = -16\)[/tex], [tex]\(b = 144\)[/tex], and [tex]\(c = 576\)[/tex]. This parabola opens downwards (since [tex]\(a\)[/tex] is negative), and the maximum height occurs at the vertex of the parabola.
The vertex of a parabola given by [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ t_{\text{max}} = -\frac{b}{2a} \][/tex]
Substituting the coefficients:
[tex]\[ t_{\text{max}} = -\frac{144}{2 \times -16} \][/tex]
[tex]\[ t_{\text{max}} = -\frac{144}{-32} \][/tex]
[tex]\[ t_{\text{max}} = 4.5 \][/tex]
To find the maximum height, we substitute [tex]\( t_{\text{max}} \)[/tex] back into the height function:
[tex]\[ h(4.5) = -16(4.5)^2 + 144(4.5) + 576 \][/tex]
[tex]\[ h(4.5) = -16(20.25) + 144(4.5) + 576 \][/tex]
[tex]\[ h(4.5) = -324 + 648 + 576 \][/tex]
[tex]\[ h(4.5) = 900 \][/tex]
Thus, the maximum height of the rocket is 900 feet, and it occurs at [tex]\( t = 4.5 \)[/tex] seconds.
### Finding the Time When the Rocket Reaches the Ground
The rocket reaches the ground when the height is zero:
[tex]\[ -16t^2 + 144t + 576 = 0 \][/tex]
This is a quadratic equation, which we can solve using the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the coefficients:
[tex]\[ a = -16 \][/tex]
[tex]\[ b = 144 \][/tex]
[tex]\[ c = 576 \][/tex]
[tex]\[ t = \frac{-144 \pm \sqrt{144^2 - 4(-16)(576)}}{2(-16)} \][/tex]
[tex]\[ t = \frac{-144 \pm \sqrt{20736 + 36864}}{-32} \][/tex]
[tex]\[ t = \frac{-144 \pm \sqrt{57600}}{-32} \][/tex]
[tex]\[ t = \frac{-144 \pm 240}{-32} \][/tex]
This gives us two solutions:
[tex]\[ t = \frac{96}{-32} = -3 \quad \text{(not physically meaningful, since time cannot be negative)} \][/tex]
[tex]\[ t = \frac{-384}{-32} = 12 \][/tex]
Thus, the rocket reaches the ground at [tex]\( t = 12 \)[/tex] seconds.
### Summary
- The rocket's maximum height is 900 feet.
- It takes 4.5 seconds for the rocket to reach its maximum height.
- The rocket reaches the ground at 12 seconds after it is fired.
These are the detailed steps needed to solve the problem correctly.
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