To find the time at which the rocket will reach its maximum height, we will analyze the given quadratic equation that models the height of the rocket. The equation is:
[tex]\[
y = -16x^2 + 111x + 118
\][/tex]
A quadratic equation of the form [tex]\(ax^2 + bx + c\)[/tex] reaches its maximum (or minimum, depending on the sign of [tex]\(a\)[/tex]) at the vertex. For a parabola that opens downwards as it does here (since the coefficient of [tex]\(x^2\)[/tex] is negative), the maximum height occurs at the vertex.
The x-coordinate of the vertex of a quadratic equation in standard form [tex]\(ax^2 + bx + c\)[/tex] can be calculated using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
For our equation, the coefficients are:
- [tex]\(a = -16\)[/tex]
- [tex]\(b = 111\)[/tex]
- [tex]\(c = 118\)[/tex] (though [tex]\(c\)[/tex] is not needed for this specific calculation)
Using the vertex formula, we plug in the values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
x = -\frac{111}{2(-16)}
\][/tex]
Simplifying the expression inside the fraction:
[tex]\[
x = -\frac{111}{-32}
\][/tex]
Since dividing by a negative number changes the sign, we get:
[tex]\[
x = \frac{111}{32}
\][/tex]
Perform the division to find the time:
[tex]\[
x \approx 3.47
\][/tex]
Thus, the rocket will reach its maximum height approximately [tex]\(3.47\)[/tex] seconds after launch.
