Question

1. (15 points) a town is infected with covid-19. the infected population increases logistically, according to the rule:

\
(t) = \frac{500}{1 + 4e^{-0.20t}}\\

where \\(t\\) is the number of days since the infection broke out.

(a) how many people are infected after 10 days? round to the nearest whole person.

ans: people

(b) how many people were initially infected? hint: the answer is not zero!

ans: people

Explanation:

Step1: Substitute \(t = 10\) into the function

$$N(10) = \frac{500}{1 + 4e^{-0.20 \times 10}}$$

Step2: Calculate the exponent value

$$-0.20 \times 10 = -2$$

Step3: Evaluate the denominator

$$1 + 4e^{-2} \approx 1 + 4 \times 0.135335 = 1.54134$$

Step4: Calculate the population at \(t = 10\)

$$N(10) \approx \frac{500}{1.54134} \approx 324.39 \approx 324$$

Step5: Substitute \(t = 0\) for initial population

$$N(0) = \frac{500}{1 + 4e^{-0.20 \times 0}}$$

Step6: Calculate the initial population

$$N(0) = \frac{500}{1 + 4 \times 1} = \frac{500}{5} = 100$$

Answer:

(a) 324 people
(b) 100 people