Calculus Roller Coaster Project Answers

Project: Designing a Roller Coaster - AP Calculus Projection: Explain...

Project: Designing a Roller Coaster - AP Calculus

Project: Explain the mathematics of roller coaster construction. Imagine you are a roller coaster designer, and are asked to mathematically correspond the curve of the first drop of a new ride. Past looking at some of your favorite coasters, you determine that the optimal slope for the rising, when the auto is being pulled up the gradient by a chain, is 0.7. The slope of the first driblet that volition be the nigh thrilling (without existence dangerous) is -1.five. You decide to connect these two straight stretches, y = L1 and y = L2 , with a parabola of the course y = f(ten) = ax two + bx + c, where x and f(x) are measured in meters. In order for the bend of the track to remain smooth, you must make sure that the linear segments, 50 1 and L2 , accept the aforementioned slope at each endpoint of the parabola, points P and Q. This ensures that the parabola is differentiable, equally well equally continuous at the endpoints. Choose point P, the left endpoint of your parabola, as your origin, to simplify calculations.

a) Describe a crude sketch of L1 with slope 0.7, ending at signal P. Then sketch L2 with slope -1.5, starting at betoken Q a short altitude to the correct and below point P. Finally, sketch the parabola that would connect P and Q while matching the slopes of L1 and L2 .

b) If the horizontal distance between P and Q is 40 meters, write equations in a, b, and c that make certain the track is shine at the connections P and Q.

c) Solve the equations for a, b, and c to find the equation of the parabola that connects P and Q.

d) Using your calculator or a figurer, plot the parabola f(x). Compare this to your original sketch in part a) past noting any similarities or differences.

e) How much higher is betoken P than bespeak Q?

f) Find the maximum elevation of the coaster algebraically. Verify your reply by graphing f(x) on your reckoner.

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