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Use the advertising awareness model described in Example 6 to find the number of people y (in millions) aware of the product as a function of time t (in years). (Round your coefficients to four decimal places.) y = 0 when t = 0; y = 0.8 when t = 5 y(t) =

Use the advertising awareness model described in Example 6 to find the number of people y (in millions) aware of the product as a function of time t (in years). (Round your coefficients to four decimal places.) y = 0 when t = 0; y = 0.8 when t = 5 y(t) =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Use the advertising awareness model described in Example 6 to find the number of people y (in millions) aware of the product as a function of time t (in years). (Round your coefficients to four decimal places.) y = 0 when t = 0; y = 0.8 when t = 5 y(t) =
Transcribed Image Text:Use the advertising awareness model described in Example 6 to find the number of people y (in millions) aware of the product as a function of time t (in years). (Round your coefficients to four decimal places.) y = 0 when t = 0; y = 0.8 when t = 5 y(t) =
EXAMPLE Modeling Advertising Awareness A new cereal product is introduced through an advertising campaign to a population of 1 million potential customers. The rate at which the population hears about the product is assumed to be proportional to the number of people who are not yet aware of the product. By the end of 1 year, half of the population has heard of the product. How many will have heard of it by the end of 2 years? Solution Let y be the number (in millions) of people at time i who have heard of the product. This means that (1 – y) is the number of people who have not heard of it, and dy/dt is the rate at which the population hears about the product. From the given assumption, you can write the differential equation as shown. = k(1 – y) dt is própor- tional to Rate of the difference change- of y between 1 and y. You can solve this equation using separation of variables. dy = k(1 – y) dt Differential form dy = k dt 1- y - In|1 - y| = kt + c, In|1 – y| = - kt – C, 1- y = e-k-C, y = 1- Ce-ki Separate variables. Integrate. Multiply cach side by - 1. Assume y < 1. General solution To solve for the constants C and k, use the initial conditions. That is, because y = 0 when i = 0, you can determine that C = 1. Similarly, because y = 0.5 when t = 1, it follows that 0.5 = 1 - e-4, which implies that k = In 2 - 0.693. So, the particular solution is y = 1- e-0.69N Particular solution This model is shown in Figure 6.15. Using the model, you can determine that the number of people who have heard of the product after 2 years is y = 1-e-0.69M2) - 0.75 or 750,000 people. 1.25 y=1-e-069 1.00- (2, 0.75) 0.75 0.50- (1,0.50) 0.25 (0, 0) Time (in years) Potential customers (in millions)
Transcribed Image Text:EXAMPLE Modeling Advertising Awareness A new cereal product is introduced through an advertising campaign to a population of 1 million potential customers. The rate at which the population hears about the product is assumed to be proportional to the number of people who are not yet aware of the product. By the end of 1 year, half of the population has heard of the product. How many will have heard of it by the end of 2 years? Solution Let y be the number (in millions) of people at time i who have heard of the product. This means that (1 – y) is the number of people who have not heard of it, and dy/dt is the rate at which the population hears about the product. From the given assumption, you can write the differential equation as shown. = k(1 – y) dt is própor- tional to Rate of the difference change- of y between 1 and y. You can solve this equation using separation of variables. dy = k(1 – y) dt Differential form dy = k dt 1- y - In|1 - y| = kt + c, In|1 – y| = - kt – C, 1- y = e-k-C, y = 1- Ce-ki Separate variables. Integrate. Multiply cach side by - 1. Assume y < 1. General solution To solve for the constants C and k, use the initial conditions. That is, because y = 0 when i = 0, you can determine that C = 1. Similarly, because y = 0.5 when t = 1, it follows that 0.5 = 1 - e-4, which implies that k = In 2 - 0.693. So, the particular solution is y = 1- e-0.69N Particular solution This model is shown in Figure 6.15. Using the model, you can determine that the number of people who have heard of the product after 2 years is y = 1-e-0.69M2) - 0.75 or 750,000 people. 1.25 y=1-e-069 1.00- (2, 0.75) 0.75 0.50- (1,0.50) 0.25 (0, 0) Time (in years) Potential customers (in millions)
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