SOLUTION: A model for the number of people N in a college community who have heard a rumor is given by the equation:{{{N=P(1-e^(-0.15d))}}} where P is the total population of the community a

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: A model for the number of people N in a college community who have heard a rumor is given by the equation:{{{N=P(1-e^(-0.15d))}}} where P is the total population of the community a      Log On


   

Question 143991: A model for the number of people N in a college community who have heard a rumor is given by the equation:N=P%281-e%5E%28-0.15d%29%29 where P is the total population of the community and d is the number of days elapsed since the rumor began. If the number of students is 1,000 find the number of days when 500 students would have heard the rumor.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
N=P%281-e%5E%28-0.15d%29%29 Start with the given equation


500=1000%281-e%5E%28-0.15d%29%29 Plug in N=500 and P=1000


500%2F1000=%281-e%5E%28-0.15d%29%29 Divide both sides by 1000


1%2F2=1-e%5E%28-0.15d%29 Reduce


1%2F2-1=-e%5E%28-0.15d%29 Subtract 1 from both sides


-1%2F2=-e%5E%28-0.15d%29 Combine like terms


%28-1%2F2%29%2F%28-1%29=e%5E%28-0.15d%29 Divide both sides by -1 (this will remove the negative sign from -e


1%2F2=e%5E%28-0.15d%29 Divide


ln%281%2F2%29=ln%28e%5E%28-0.15d%29%29 Take the natural log of both sides


ln%281%2F2%29=ln%28e%5E%28-0.15d%29%29 Take the natural log of both sides


ln%281%2F2%29=-0.15d%2Aln%28e%29 Rewrite the expression using the identity ln%28x%5Ey%29=y%2Aln%28x%29%29


ln%281%2F2%29=-0.15d%281%29 Take the natural log of "e" to get 1


ln%281%2F2%29=-0.15d Multiply


ln%281%2F2%29%2F%28-0.15%29=d Divide both sides by -0.15 to isolate d



So our answer is d=-ln%281%2F2%29%2F%280.15%29 which is approximately d=4.62098


So in a little more than 4 and half days, 500 students will have heard the rumor