Question

A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find out the time at which the rocket will reach its max, to the nearest 100th of a second.

y = - 16x ^ 2 + 193x + 123

Answer 1

To find the time at which the rocket reaches its maximum height, we need to determine the vertex of the parabolic equation given. The equation of the height of the rocket is:

\[
y = -16x^2 + 193x + 123
\]

For a quadratic equation in the form \(y = ax^2 + bx + c\), the x-coordinate of the vertex (which corresponds to the maximum or minimum of the parabola) can be calculated using the formula:

\[
x = -\frac{b}{2a}
\]

In this equation, \(a = -16\) and \(b = 193\). Plugging in these values:

\[
x = -\frac{193}{2 \times -16} = -\frac{193}{-32} = \frac{193}{32}
\]

Now, we calculate \( \frac{193}{32} \):

\[
\frac{193}{32} = 6.03125
\]

Now rounding this to the nearest hundredth (two decimal places):

\[
x \approx 6.03
\]

Thus, the rocket will reach its maximum height at approximately **6.03 seconds** after launch.