Question

Roller C...

M
DED
M
The goal of this project is to design a roller coaster and compute its thrill.
Definitions:
• A drop of a roller coaster is defined as an interval for which the function is strictly decreasing
• The angle of descent at a point is defined as the angle between the horizontal and the line
tangent to the function at the drop
•
The thrill of a drop is defined as the product between the angle of steepest descent during the
drop and the total vertical distance of the drop
• The thrill of a roller coaster is defined as the sum of the thrills in each drop of the roller coaster
Limitations:
A roller coaster is the graph of a function r(x) with domain [0, 200] such that:
the roller coaster starts on the ground r(0) = 0
the maximum height of the roller coaster is 75 meters: r(x) ≤ 75 for all x = [0, 200]
• the roller coaster does not go below 25 meters underground: r(x) 2 -25 for all x = [0, 200]
the ride is smooth: r(x) is differentiable everywhere on its domain
the angle of steepest descent for the roller coaster is never more than 90 degrees

Answer 1

Explanation:

To design a roller coaster and compute its thrill, we need to follow the given definitions and limitations. Here's a step-by-step approach:

Design the Roller Coaster Function:

We need to design a differentiable function that represents the shape of the roller coaster. Let's denote this function as r(x), where x is the horizontal distance.

The function should satisfy the given limitations: r(0) = 0 (start on the ground), r(x) ≤ 75 (maximum height), and r(x) ≥ -25 (above the underground).

The function should be differentiable over the interval [0, 200] to ensure a smooth ride.

Identify Drops:

Drops occur where the function is strictly decreasing. We can find these drops by analyzing the intervals where r'(x) < 0 (negative slope).

Each drop will be an interval with a start and end point.

Compute Angle of Descent:

To calculate the angle of descent at a point on the drop, we need to find the tangent line to the function at that point.

The angle of descent is the angle between the horizontal line and the tangent line.

We can use the derivative of the function, r'(x), to find the slope of the tangent line.

The angle can be calculated using trigonometry: angle = arctan(r'(x)).

Calculate Thrill of Each Drop:

The thrill of a drop is the product of the angle of steepest descent during the drop and the total vertical distance of the drop.

The vertical distance of a drop is the difference between the function values at the start and end points of the drop.

Calculate the angle of steepest descent for each drop and multiply it by the vertical distance to obtain the thrill of that drop.

Compute Total Thrill of the Roller Coaster:

The total thrill of the roller coaster is the sum of the thrills of all the drops.

Add up the individual thrill values calculated for each drop to get the overall thrill of the roller coaster.

Please note that the actual implementation of these steps requires specific mathematical calculations and programming. If you need assistance with any particular step or have further questions, feel free to ask.