(a) Suppose that Ak, Bk, and Ck are the percentages of the Cheggers fleet at each of the three lots (Airport, Broadway and Calvin, respectively) at the beginning of day k (before any cars are rented for the day). Find the system of linear equations that describes how the percentages at the beginning of day k+1, Ak+1, Bk+1, and Ck+1, depend on the percentages on day k, Ak, Bk, and Ck, and express this relationship in the form Fk+1 = M Fk, where M is a 3 × 3 matrix and Fk = Ak Вк Ck (b) Explain (briefly) why λ = 1 is an eigenvalue of M, and find an eigenvector of M with this eigenvalue. Show your work, including at least two intermediate matrices in the Gaussian elimination process. (c) Find the other eigenvalue(s) of M. You do not need to find corresponding eigenvector(s). On January 1st, 2023, before any cars are rented, Ao = 50%, Bo = 25%, and Co = 25%; (d) Based on your answer to part (b), estimate the percentages of cars in each of the three lots at the beginning of January 1, 2024. Explain the reasoning, including why your work on part (c) is not important for this estimate.
(a) Suppose that Ak, Bk, and Ck are the percentages of the Cheggers fleet at each of the three lots (Airport, Broadway and Calvin, respectively) at the beginning of day k (before any cars are rented for the day). Find the system of linear equations that describes how the percentages at the beginning of day k+1, Ak+1, Bk+1, and Ck+1, depend on the percentages on day k, Ak, Bk, and Ck, and express this relationship in the form Fk+1 = M Fk, where M is a 3 × 3 matrix and Fk = Ak Вк Ck (b) Explain (briefly) why λ = 1 is an eigenvalue of M, and find an eigenvector of M with this eigenvalue. Show your work, including at least two intermediate matrices in the Gaussian elimination process. (c) Find the other eigenvalue(s) of M. You do not need to find corresponding eigenvector(s). On January 1st, 2023, before any cars are rented, Ao = 50%, Bo = 25%, and Co = 25%; (d) Based on your answer to part (b), estimate the percentages of cars in each of the three lots at the beginning of January 1, 2024. Explain the reasoning, including why your work on part (c) is not important for this estimate.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Cheggers Cars rents cars in a large, East Coast city, with an
unusual business plan: every car rented from one of Cheggers'
three lots - the Airport lot, the Broadway lot and the Calvin
Street lot - must be returned at the end of the day to one of
these lots.
The management of Cheggers Cars has observed the
following daily patterns:
1. Of the cars rented from the Airport lot:
• 30% are returned to the Airport lot
there that are not rented that day.
●
• 30% are returned to the Broadway lot.
●
• 40% are returned to the Calvin St. lot.
2. Of the cars rented from the Broadway lot:
• 40% are returned to the Airport lot.
40% are returned to the Broadway lot including the cars
there that are not rented that day.
• 20% are returned to the Calvin St. lot.
●
including the cars
3. Of the cars rented from the Calvin Street lot:
• 50% are returned to the Airport lot.
30% are returned to the Broadway lot.
20% are returned to the Calvin St. lot including the cars
there that are not rented that day.
●
●
Transcribed Image Text:(a) Suppose that Ak, Bk, and Ck are the percentages of
the Cheggers fleet at each of the three lots (Airport, Broadway
and Calvin, respectively) at the beginning of day k (before any
cars are rented for the day). Find the system of linear
equations that describes how the percentages at the beginning
of day k+1, Ak+1, Bk+1, and Ck+1, depend on the percentages
on day k, Ak, Bk, and Ck, and express this relationship in the
form Fk+1 = M Fk, where M is a 3 x 3 matrix and
Fk
=
Ак
B
Ck
(b)
Explain (briefly) why λ = 1 is an eigenvalue of M,
and find an eigenvector of M with this eigenvalue. Show your
work, including at least two intermediate matrices in the
Gaussian elimination process.
(c)
Find the other eigenvalue(s) of M. You do not need
to find corresponding eigenvector(s).
On January 1st, 2023, before any cars are rented,
Ao 50%, Bo = 25%, and Co = 25%;
=
(d)
Based on your answer to part (b), estimate the
percentages of cars in each of the three lots at the beginning of
January 1, 2024. Explain the reasoning, including why your
work on part (c) is not important for this estimate.
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