Engineer: Roller Coaster

The Roller Coaster Design Challenge 🎢

An amusement park is designing a new roller coaster with a parabolic track. The engineers are testing a prototype where the height of the coaster (in meters) above the ground is modeled by the equation:

where h(t) represents the height of the roller coaster at time t seconds after it starts descending from its highest point.

Open-Ended Inquiry Questions:

1. Interpret the equation: What do the numbers in the equation tell us about the motion of the roller coaster? Explain in terms of the real-world context.
   
   

2. Critical Thinking: At what time does the roller coaster reach the ground? How could you confirm your answer using different methods (graphing, algebraic solving, or estimation)?
   
   

3. Design Extension: Suppose the amusement park wants to make the ride more thrilling by increasing the highest point to 30 meters while keeping the same shape. How would you modify the equation?
   
   

4. Optimization & Real-World Application: If the engineers want to ensure that the coaster does not go above a certain speed for safety reasons, what other mathematical factors (besides height) should they consider in their design? How might you model this mathematically? At what time(s), t, will the roller coaster have the highest speed(s)? At what time(s), t, will the roller coaster have the slowest speed(s)? At what time, t, will the roller coaster have a speed of zero? Do your answers make sense in real life?