Question

The middle school portion of a roller coaster truck travels along a parabolic path when the roller coaster is at the vertex of the path the focus is 25 feet above it. write am equation that represents the path of the roller coaster. assume that the focus is on the positive y axis and the vertex is 0,0 when the width of the parabolic track is 20 feet what is the vertical height of the roller coaster in relation to the vertex. what is the width of the parabolic track when the roller coaster is at a height of 25 feet in relation to the vertex

Answer 1

The equation representing the path of a roller coaster with its vertex at the origin and focus 25 feet above it is y = x^2/100. The vertical height at a track width of 20 feet is 1 foot. When the roller coaster is at a height of 25 feet, the track width is 50 feet.

Step-by-step explanation:

To represent the path of the roller coaster with the vertex at (0,0) and the focus 25 feet above it, we use the standard form of a parabolic equation: y = ax^2. Since the focus is 25 feet above the vertex, we can determine 'a' using the equation 1/(4a) = 25, which gives us a = 1/100. Thus, the equation representing the path is y = x^2/100.

For a parabolic track with a width of 20 feet, the equation tells us that this width occurs at a certain height 'h'. Setting this distance as the 'x' value and solving for 'y', we find that y = (10)^2/100 = 1 foot. Therefore, the vertical height of the roller coaster in relation to the vertex when the width is 20 feet is 1 foot.

Conversely, to find the width when the roller coaster is at a height of 25 feet, we set 'y' to 25 in the equation and solve for 'x' to get x = ±sqrt(2500), which results in a width of 50 feet (the distance from 'x' to '-x').