the number of people in a small town who are reached by a rumor about the mayor and an intern is given by n…

the number of people in a small town who are reached by a rumor about the mayor and an intern is given by n = 30,000 / (1 + 300e^(-0.6t)), where t is the number of days after the rumor begins. complete parts (a)-(c). 181 (round to the nearest integer as needed.) b) how many will have heard the rumor by the end of the fifth day? 1883 (round to the nearest integer as needed.) c) use graphical or numerical methods to find the day on which 7800 people in town have heard the rumor. th day (round up to the nearest integer.)

Answer

Explanation:

Step1: Set up the equation

We are given the formula $N=\frac{30000}{1 + 300e^{-0.6t}}$ and we want to find $t$ when $N = 7800$. So we set up the equation $7800=\frac{30000}{1 + 300e^{-0.6t}}$.

Step2: Cross - multiply

Cross - multiplying gives us $7800(1 + 300e^{-0.6t})=30000$. Then $1 + 300e^{-0.6t}=\frac{30000}{7800}=\frac{50}{13}$.

Step3: Isolate the exponential term

Subtract 1 from both sides: $300e^{-0.6t}=\frac{50}{13}-1=\frac{50 - 13}{13}=\frac{37}{13}$. Then $e^{-0.6t}=\frac{37}{13\times300}=\frac{37}{3900}$.

Step4: Take the natural logarithm

Taking the natural logarithm of both sides, we get $\ln(e^{-0.6t})=\ln(\frac{37}{3900})$. Using the property $\ln(e^x)=x$, we have $- 0.6t=\ln(37)-\ln(3900)$.

Step5: Solve for t

$t=-\frac{\ln(37)-\ln(3900)}{0.6}$. Calculate $\ln(37)\approx3.6109$, $\ln(3900)\approx8.2674$. Then $t=-\frac{3.6109 - 8.2674}{0.6}=-\frac{-4.6565}{0.6}\approx7.76$.