A rocket is launched from a tower. The height of the rocket, [tex]\( y \)[/tex] in feet, is related to the time after launch, [tex]\( x \)[/tex] in seconds, by the given equation. Using this equation, find the time the rocket will reach its maximum height, to the nearest hundredth of a second.

[tex]\[ y = -16x^2 + 111x + 118 \][/tex]



Answer :

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.