You have been hired by a firm that is designing a runway for an airport. Your job is to confirm that the runway is long enough for an airplane to land, but not unnecessarily long. You are given the velocity data for the largest aircraft that will land at the airport. The velocity data will be used to calculate the distance required for the aircraft to safely land and come to a stop. Below is a set of data that represents the velocity (in mph) of the final 45 seconds of the landing. At t = 0, the plane is on its final descent.
Table I t in seconds 0 5 10 15 20 25 30 35 40 45
v(t) in mph18715 2.1812 2.296.4 1 74.21 55.1 38.64 24.49 12.3 1.81
Table IIt in seconds 9 10 19 20 29 30 39 40 44 45
v(t) in mph 127.84 122.2 78.38 74.21 41.74 38.64 14.6 12.3 3.79 1.81
Part I: Introduction
Describe the purpose and problem statement of this report. How do you think you will use calculus to ensure that the runway is sufficient? How are the velocity data useful?
Part II: Analysis of Data (Derivatives)
Using the data in Table I, calculate the average acceleration for the following intervals:
From t = 0 to t = 45
From t = 25 to t = 45
From t = 40 to t = 45
Using the data in Table II, calculate the instantaneous acceleration at t = 10, t = 20, t = 30, t = 40, and t = 45 seconds.
Explain how you used the limit definition of a derivative to calculate the instantaneous acceleration. Use your results to explain why the limit definition of a derivative is true.
At what point is the acceleration at a maximum? How is this relevant to the landing aircraft?
For each of the questions above, provide supporting solutions and calculus terminology to explain how you arrived at your answers. Also, explain in detail what your answer represents in a real-world context and why this is useful.
Part III: Analysis of Data (Integrals)
Using the data in Table I, use a right-endpoint estimate to calculate the distance traveled by the aircraft from t = 0 to t = 45 seconds.
Using the data in Table I, use a left-endpoint estimate to calculate the distance (total change) traveled by the aircraft from t = 0 to t = 45 seconds.
Find the best estimate for the distance traveled by the aircraft from t = 0 to t = 45 seconds.
Using the velocity data in Table I, generate a table of estimates that represents the total change of the moving object:
t in seconds051015202530354045
v(t) in mph187152.18122.296.4174.2155.138.6 424.4912.31.81
s(t)0
Part IV: Analysis of the Model: Calculate parts I and II above.
The model for these sets of data is given by v(t) = 250e^-0.03t - 63 Use this model to calculate the instantaneous acceleration at t = 15, t = 25, t = 35, and t = 45 seconds.
Compare the results from parts I and II to the results found in parts III. A and III. B. Discuss which method is more accurate and explain the relevance of each in a real-world context.
Part V: Final Results and Recommendations: In this section, you will conclude with your recommendations from your findings.
Compare the results obtained in parts II and III with the results in part IV. Discuss the accuracy of each method, and explain the application of each method in a real-world context.
Defend your process by identifying the appropriate explanation for each process step.