Respuesta :
a) it will take 31 days
b) it will take 54 days
Explanation:
The number of people who have been exposed to a news item in a city after t days is given as:
[tex]p\mleft(t\mright)=2,800,000\mleft(1-e^{-0.03t}\mright)[/tex]a) We need to find the number of days it took 60% of the population to hear the news
p(t) = 60% of the population
Total population = 2.8 million
p(t) = 60% × 2800000 = 1680000
t = ?
substitute the value of p(t) to get t:
[tex]\begin{gathered} 1680000=2800000\mleft(1-e^{-0.03t}\mright) \\ \text{divide both sides by 2800000}\colon \\ \frac{1680000}{\text{2800000}}=1-e^{-0.03t} \\ 0.6\text{ = }1-e^{-0.03t} \\ 0.6\text{ - 1 = }-e^{-0.03t} \\ -0.4\text{ = }-e^{-0.03t} \\ \\ \text{divide both sides by -1:} \\ 0.4\text{ = }e^{-0.03t} \end{gathered}[/tex][tex]\begin{gathered} \text{Taking natural log of both sides:} \\ \log _e0.4\text{ =}\log _e(\text{ }e^{-0.03t}) \\ \log _e0.4\text{ = -0.03t} \\ \ln \text{ 0.4 = -0.03t} \\ \\ \text{divide both sides by -0.03:} \\ \frac{\ln\text{ 0.4}}{\text{-0.03}}\text{ = t} \\ t\text{ = }30.5430\text{ } \\ \\ To\text{ the nearest day, it will take 31 days} \end{gathered}[/tex]b) We need to find the number of days it took 80% of the population to hear the news
p(t) = 80% of the population = 80% (2800000)
p(t) = 2240000
substitute the value of p(t) into the formula:
[tex]\begin{gathered} 2240000=2800000\mleft(1-e^{-0.03t}\mright) \\ \text{divide both sides by 2800000:} \\ \frac{2240000}{2800000}\text{ = }1-e^{-0.03t} \\ 0.8\text{ = }1-e^{-0.03t} \\ 0.8\text{ + }e^{-0.03t}\text{ = 1} \\ e^{-0.03t}\text{ = 1 - 0.8} \\ e^{-0.03t}\text{ = 0.2} \end{gathered}[/tex][tex]\begin{gathered} \text{Taking natural log of both sides:} \\ \ln (e^{-0.03t})\text{ = ln 0.2} \\ -0.03t\text{ = ln 0.2} \\ \text{divide both sides by -0.03:} \\ t\text{ = }\frac{\ln \text{ 0.2}}{-0.03} \\ t\text{ = 53.6479} \\ \\ To\text{ the nearest days, it will take 54 days} \end{gathered}[/tex]
