A woman is listening to her radio, which is 174 m from the radio station transmitter. (a) How many wavelengths of the radio waves are there between the transmitter and radio receiver if the woman is listening to an AM radio station broadcasting at 1260 kHz ? wavelengths (b) How many wavelengths are there between the transmitter and radio if she changes to an FM station broadcasting at 101.7 MHz? wavelengths
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/6372428458LgmZXNxqka.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/6896791134NPlCNIwqTC.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/1401586756DoSHMhHJpq.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/9702922078OiTwyPXqkE.png)
![Definite integrals are useful to science because we can often describe natural phenomena using definite integrals. However, integrating functions arising in real-life science can be rather challenging. In this exercise, we see how having only the graph of some functions is enough to answer some questions in science. The physical interpretation of an integral is a prerequisite knowledge you learned in MATH 1013. For a refresher, see the Net Change Theorem in Section 5.4 of the textbook. A group of researchers studies the population of arctic foxes in Northern Canada. Initially, the researchers identified a population of 1,000 arctic foxes. Due to various factors, the population of arctic foxes changes over time. The graph of the birthrate B(t) and the death rate D(t), for the population, are shown below. Time t is measured in years. (1) In a sentence or two, describe the meaning of the function f(t) = t 0 B(x) - D(x)dx in the context of this problem. (2) According the graph, is there a time t between 0 and 20 at which the population is equal to the initial population? If so, provide a rough estimate of this time, and write one or two sentences explaining how you found it. Your estimate should be a single number, not an interval. (3) Recall from Quiz 1 the definition of the average value of a function f(x). The average value of a function f(x) over an interval [a, b] is denoted by favg and is defined to be favg = 1 b - a b a f(x)dx. Suppose you know that the average birth rate over the first 20 years is 90 arctic foxes per year and the average death rate over the first 20 years is 100 arctic foxes per year. What is the population of the arctic foxes at t = 20 ? Show your work.](https://www.doubtrix.com/uploads/editor/1288831183fhExDQGbmw.png)
