Question

What are the coordinates of the maximum point defined by the equation f(x) = 38 + 27x - 0.512x², where h represents the height in kilometers and t represents the time in minutes for the path of a ballistic missile tracked on a computer screen in Algebra 2? Additionally, what significance or meaning do these coordinates hold in the context of the ballistic missile's trajectory?

Answer 1

The coordinates of the maximum point are (26.37, 905.59). These coordinates represent the time and height at which the missile reaches its maximum height on the ballistic trajectory.

Step-by-step explanation:

The equation f(x) = 38 + 27x - 0.512x² represents the height of a ballistic missile as a function of time. To find the coordinates of the maximum point, we need to determine the vertex of the parabolic function. The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a, b, and c are the coefficients of the quadratic equation. In this case, a = -0.512 and b = 27. Substituting these values into the formula gives x = -27/(2*(-0.512)) = 26.3671875.

To find the corresponding y-coordinate, we substitute this x-value back into the function. So, f(26.3671875) = 38 + 27(26.3671875) - 0.512(26.3671875)² = 905.59375. Therefore, the coordinates of the maximum point are (26.3671875, 905.59375).

In the context of the ballistic missile's trajectory, these coordinates represent the time and height at which the missile reaches its maximum height. The x-coordinate represents the time, and the y-coordinate represents the height in kilometers. So, at approximately 26.37 minutes after launch, the missile reaches a maximum height of approximately 905.59 kilometers.