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Answer :
To determine how long it will take for the toy rocket to reach its maximum height, we need to analyze the height equation given for the rocket:
[tex]\[ h(t) = -16t^2 + 128t \][/tex]
This equation represents a parabola that opens downward because the coefficient of [tex]\( t^2 \)[/tex] is negative. The maximum height is achieved at the vertex of this parabola.
For a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], the time [tex]\( t \)[/tex] at which the vertex (and hence the maximum or minimum value) occurs is given by the formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
In our equation:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 128 \)[/tex]
Now, plug these values into the formula:
[tex]\[ t = -\frac{128}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{128}{-32} \][/tex]
[tex]\[ t = 4 \][/tex]
So, it will take the rocket 4 seconds to reach its maximum height.
[tex]\[ h(t) = -16t^2 + 128t \][/tex]
This equation represents a parabola that opens downward because the coefficient of [tex]\( t^2 \)[/tex] is negative. The maximum height is achieved at the vertex of this parabola.
For a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], the time [tex]\( t \)[/tex] at which the vertex (and hence the maximum or minimum value) occurs is given by the formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
In our equation:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 128 \)[/tex]
Now, plug these values into the formula:
[tex]\[ t = -\frac{128}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{128}{-32} \][/tex]
[tex]\[ t = 4 \][/tex]
So, it will take the rocket 4 seconds to reach its maximum height.
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