Roller Coaster Route?

📌 CONCEPT: The problem involves finding the slope of a line that represents the roller coaster track, given an angle and a vertical displacement.

📐 RULE / FORMULA: The slope of a line is given by the tangent of the angle it makes with the horizontal, which is expressed as tan(θ) = rise / run. In this case, we can use the angle of 30° to find the slope of the track.

💡 WORKED EXAMPLE: Let's say the starting point is at sea level, and the peak of the hill is 30° above the horizontal. If the first drop is 5 meters below the starting point, the track's slope will change from 30° to an angle that has a tangent equal to 5 meters (the rise). We can find this angle by taking the inverse tangent of 5 meters and dividing it by the horizontal distance between the starting point and the drop.

⚠️ COMMON MISTAKE: Students often forget to consider the horizontal distance when finding the new slope after the drop, which can lead to incorrect calculations.