George and Sandra both decide to open small cafes in town, on the same block: "George's Gourmet Tea" and "Sandra's Super Coffee." They soon become concerned about their effects on each other. Will having two beverage shops so close make them compete for a limited number of customers, or will it make their block a more popular destination and so attract more customers? Let x(t)represent the daily profit (scaled to have an appropriate unit, like hundreds of dollars) at George's cafe at time t, and let y(t) represent the daily profit at Sandra's cafe at time t. Note that x and y can take values that are positive (profit), negative (loss), and zero (break-even). We will model the rates of change of their daily profits with the linear system dx/dt=-x+2y dy/dt=-2x+y (a) Interpret the terms of the system and how they relate to the rates of change of their daily profits. In other words, what does this model assume is true about the way daily profit at one cafe affects its own and the other's daily profit rate of change? Explain thoroughly. (b) Solve the IVP (initial value problem) comprised of the system above and the initial condition (x(0), y(0))=(2,2). Show your work and/or explain your process. (c) Describe the long-term predictions that the model makes for x and y. Interpret these predictions in the context of George and Sandra's cafes.
George and Sandra both decide to open small cafes in town, on the same block: "George's Gourmet Tea" and "Sandra's Super Coffee." They soon become concerned about their effects on each other. Will having two beverage shops so close make them compete for a limited number of customers, or will it make their block a more popular destination and so attract more customers?
Let x(t)represent the daily profit (scaled to have an appropriate unit, like hundreds of dollars) at George's cafe at time t, and let y(t) represent the daily profit at Sandra's cafe at time t. Note that x and y can take values that are positive (profit), negative (loss), and zero (break-even). We will model the rates of change of their daily profits with the linear system
(a) Interpret the terms of the system and how they relate to the rates of change of their daily profits. In other words, what does this model assume is true about the way daily profit at one cafe affects its own and the other's daily profit rate of change? Explain thoroughly.
(b) Solve the IVP (initial value problem) comprised of the system above and the initial condition (x(0), y(0))=(2,2). Show your work and/or explain your process.
(c) Describe the long-term predictions that the model makes for x and y. Interpret these predictions in the context of George and Sandra's cafes.
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