Question

The curve part of the track of the roller coaster is represented by an equation, y = 1/64x^3 - 3/16x^2 with point P as the origin. Find the shortest vertical distance, in m, from the track to ground level.​

Answer 1

The shortest vertical distance from the roller coaster track with the given equation to ground level at the origin (point P) is 0 meters since point P itself is on the ground level.

Step-by-step explanation:

To find the shortest vertical distance from the track to ground level, you need to determine the minimum value of the function y = (1/64)x^3 - (3/16)x^2. Since the given equation represents the curve part of the roller coaster track, finding the minimum value will give you the shortest vertical distance.

The given equation is a cubic function, and its minimum value occurs at the critical points. To find the critical points, take the derivative of the function and set it equal to zero.

This equation has two solutions: x = 0 and x = 24.

Now, evaluate the function at these critical points and at the endpoints of the roller coaster track (which are not given, but let's assume x approaches infinity in one direction).

The minimum value is the lowest point among these values. Therefore, the shortest vertical distance from the track to ground level occurs at the origin, which is y=0.

Hence, the shortest vertical distance is 0 m.