Q:

A small town has 800 inhabitants. At 8 AM, 120 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Do not round k in your calculation. Round the final answer to one decimal place.)

Accepted Solution

A:
Answer:At what time will 90% of the population have heard the rumor?The rumor was heard by 90% of the population in 5.9 hours or 5 hours and 54 minutesSo, at 1:54 pm, 90% of the population had heard the rumor.Step-by-step explanation:To resolve this exercise we need to know the exponential model:[tex]P_(_t_)= P_0 *e^k^t[/tex] (1)Where:[tex]P_(_t_):[/tex] the quantity inhabitants in certain time who heard the rumor[tex]P_0:[/tex] Initial people who heard the rumork: constantt: time frameWe know in 4 hours ([tex]12-8=4[/tex] hours) half the town has heard the rumor because:[tex]Half town= \frac{800}{2}=400[/tex] inhabitantsWith this information we can find the constant (k), because we have all the information in [tex]t=4 hours[/tex][tex]P_(_4_)=[/tex] 400 people[tex]P_0=[/tex] 120 peoplet= 4 hoursWhen we replace in equation 1 we have:[tex]400=120e^4^k[/tex][tex]\frac{400}{120} =e^4^k[/tex][tex]\frac{10}{3}=e^4^k[/tex]We multiply by natural logarithm on both sides of this equation and we have:[tex]Ln(\frac{10}{3} )= 4*k\\k=\frac{Ln(\frac{10}{3} )}{4}[/tex]With the constant (k) we can find at what time 90% of the population have heard the rumor[tex]800*0.9=720 people[/tex] (90% of the population)So we have:[tex]P_(_t_)=[/tex] 720 people[tex]P_0=[/tex] 120 people[tex]k=\frac{Ln\frac{10}{3}}{4}[/tex]When we replace in equation 1 we have:[tex]720=120*e^\frac{Ln\frac{10}{3} }{4} ^*^t[/tex][tex]\frac{720}{120}=e^\frac{Ln\frac{10}{3}}{4}^t\\6=e^\frac{Ln\frac{10}{3}}{4}^t[/tex]We multiply by natural logarithm on both sides of this equation and we have:[tex]Ln 6 = \frac{Ln\frac{10}{3}}{4}*t\\[/tex] Β [tex]4*Ln 6 = Ln\frac{10}{3}*t\\t=\frac{4*Ln 6 }{Ln\frac{10}{3}}[/tex][tex]t=\frac{4*1.79}{1.20} \\t=\frac{7.16}{1.20} \\t=5.9 hours[/tex]We can find how many minutes are 0.9 hours:[tex]0.9hours\frac{60 minutes}{1hour} = 54 minutes[/tex]t= 5 hours and 54 minutesNow, we know the rumor was heard by 90% of the population in 5.9 hours or 5 hours and 54 minutesSo, at 1:54 pm, 90% of the population had heard the rumor.