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Answer :
To find out how long it takes for the toy rocket to reach its maximum height and what that height is, we need to analyze the height function of the rocket:
[tex]\[ h(t) = -16t^2 + 48t + 9 \][/tex]
This function is a quadratic equation, where:
- The coefficient of [tex]\(t^2\)[/tex] is [tex]\(a = -16\)[/tex],
- The coefficient of [tex]\(t\)[/tex] is [tex]\(b = 48\)[/tex],
- The constant term is [tex]\(c = 9\)[/tex].
### Finding the Time of Maximum Height
For any quadratic function in the form [tex]\(ax^2 + bx + c\)[/tex], the maximum or minimum value occurs at [tex]\(t = -\frac{b}{2a}\)[/tex]. This is known as the vertex formula for parabolas.
Plugging in our values:
- [tex]\(a = -16\)[/tex],
- [tex]\(b = 48\)[/tex].
The time [tex]\(t\)[/tex] when the rocket reaches its maximum height is:
[tex]\[ t = -\frac{48}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
So, it takes 1.5 seconds for the rocket to reach its maximum height.
### Calculating the Maximum Height
Once we have the time at which the maximum height occurs, we substitute [tex]\(t = 1.5\)[/tex] back into the height function to find the maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 9 \][/tex]
[tex]\[ h(1.5) = -16(2.25) + 72 + 9 \][/tex]
[tex]\[ h(1.5) = -36 + 72 + 9 \][/tex]
[tex]\[ h(1.5) = 45 \][/tex]
Therefore, the maximum height reached by the rocket is 45 feet.
In summary:
- The rocket reaches its maximum height after 1.5 seconds.
- The maximum height of the rocket is 45 feet.
[tex]\[ h(t) = -16t^2 + 48t + 9 \][/tex]
This function is a quadratic equation, where:
- The coefficient of [tex]\(t^2\)[/tex] is [tex]\(a = -16\)[/tex],
- The coefficient of [tex]\(t\)[/tex] is [tex]\(b = 48\)[/tex],
- The constant term is [tex]\(c = 9\)[/tex].
### Finding the Time of Maximum Height
For any quadratic function in the form [tex]\(ax^2 + bx + c\)[/tex], the maximum or minimum value occurs at [tex]\(t = -\frac{b}{2a}\)[/tex]. This is known as the vertex formula for parabolas.
Plugging in our values:
- [tex]\(a = -16\)[/tex],
- [tex]\(b = 48\)[/tex].
The time [tex]\(t\)[/tex] when the rocket reaches its maximum height is:
[tex]\[ t = -\frac{48}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
So, it takes 1.5 seconds for the rocket to reach its maximum height.
### Calculating the Maximum Height
Once we have the time at which the maximum height occurs, we substitute [tex]\(t = 1.5\)[/tex] back into the height function to find the maximum height:
[tex]\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 9 \][/tex]
[tex]\[ h(1.5) = -16(2.25) + 72 + 9 \][/tex]
[tex]\[ h(1.5) = -36 + 72 + 9 \][/tex]
[tex]\[ h(1.5) = 45 \][/tex]
Therefore, the maximum height reached by the rocket is 45 feet.
In summary:
- The rocket reaches its maximum height after 1.5 seconds.
- The maximum height of the rocket is 45 feet.
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