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 = 5y(t) = EXAMPLEModeling Advertising AwarenessA new cereal product is introduced through an advertising campaign to a population of1 million potential customers. The rate at which the population hears about the productis assumed to be proportional to the number of people who are not yet aware of theproduct. By the end of 1 year, half of the population has heard of the product. Howmany 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 theproduct. This means that (1 – y) is the number of people who have not heard of it, anddy/dt is the rate at which the population hears about the product. From the givenassumption, you can write the differential equation as shown.= k(1 – y)dtis própor-tional toRate ofthe differencechange-of ybetween 1and y.You can solve this equation using separation of variables.dy = k(1 – y) dtDifferential formdy= k dt1- y- In|1 - y| = kt + c,In|1 – y| = - kt – C,1- y = e-k-C,y = 1- Ce-kiSeparate variables.Integrate.Multiply cach side by - 1.Assume y < 1.General solutionTo solve for the constants C and k, use the initial conditions. That is, because y = 0when i = 0, you can determine that C = 1. Similarly, because y = 0.5 when t = 1, itfollows that 0.5 = 1 - e-4, which implies thatk = In 2 - 0.693.So, the particular solution isy = 1- e-0.69NParticular solutionThis model is shown in Figure 6.15. Using the model, you can determine that thenumber of people who have heard of the product after 2 years isy = 1-e-0.69M2)- 0.75 or 750,000 people.1.25y=1-e-0691.00-(2, 0.75)0.750.50-(1,0.50)0.25(0, 0)Time (in years)Potential customers (in millions)

From the given example, the function that models the number of people y (in millions) aware of the product as a function of time t (in years) is y=1-Ce-kt.Now find the constants C and k using the given initial conditions as follows. When t=0, we have y=0. yt=1-Ce-kty0=1-Ce-k(0)1-Ce0=0C=1 Therefore, yt=1-e-kt. Using the condition y5=0.8 , find the constant k as shown below. yt=1-e-kty5=1-e-5k0.8=1-e-5ke-5k=0.2-5k=ln0.2k=-15ln0.2 ≈0.3219Hence, the function that models the number of people y (in millions) aware of the product as a function of time t (in years) is y(t)=1-e-0.3219t.