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Answer :
To find the rocket's maximum height and the time it takes to reach that height, we need to analyze the given quadratic equation for the height of the rocket:
[tex]\[ h = -16t^2 + 112t + 704 \][/tex]
This equation is a quadratic in the form [tex]\( at^2 + bt + c \)[/tex], where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 112 \)[/tex]
- [tex]\( c = 704 \)[/tex]
The vertex of a parabola (which in this case opens downward since [tex]\( a < 0 \)[/tex]), gives the maximum point when the parabola opens downwards. The formula to find the time, [tex]\( t \)[/tex], at which the maximum height occurs is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into this formula, we get:
[tex]\[ t = -\frac{112}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{112}{-32} \][/tex]
[tex]\[ t = 3.5 \][/tex]
Therefore, it takes [tex]\( 3.5 \)[/tex] seconds for the rocket to reach its maximum height.
Now, to find the maximum height, we substitute [tex]\( t = 3.5 \)[/tex] back into the original height equation:
[tex]\[ h = -16(3.5)^2 + 112(3.5) + 704 \][/tex]
This simplification gives a maximum height of:
[tex]\[ h = 900 \][/tex]
Thus, the rocket's maximum height is 900 feet, and it takes [tex]\( 3.5 \)[/tex] seconds to reach this height.
[tex]\[ h = -16t^2 + 112t + 704 \][/tex]
This equation is a quadratic in the form [tex]\( at^2 + bt + c \)[/tex], where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 112 \)[/tex]
- [tex]\( c = 704 \)[/tex]
The vertex of a parabola (which in this case opens downward since [tex]\( a < 0 \)[/tex]), gives the maximum point when the parabola opens downwards. The formula to find the time, [tex]\( t \)[/tex], at which the maximum height occurs is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into this formula, we get:
[tex]\[ t = -\frac{112}{2 \times -16} \][/tex]
[tex]\[ t = -\frac{112}{-32} \][/tex]
[tex]\[ t = 3.5 \][/tex]
Therefore, it takes [tex]\( 3.5 \)[/tex] seconds for the rocket to reach its maximum height.
Now, to find the maximum height, we substitute [tex]\( t = 3.5 \)[/tex] back into the original height equation:
[tex]\[ h = -16(3.5)^2 + 112(3.5) + 704 \][/tex]
This simplification gives a maximum height of:
[tex]\[ h = 900 \][/tex]
Thus, the rocket's maximum height is 900 feet, and it takes [tex]\( 3.5 \)[/tex] seconds to reach this height.
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