Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 26, 2023

Find the initial number of people ill with the flu, given the function $N(t)=\frac{5000}{1+999 e^{-t}}$ models the number of people ill $t$ weeks after the outbreak. Round the result to the nearest person.

STEP 1

Assumptions1. The function $(t)=\frac{5000}{1+999 e^{-t}}$ models the number of people who have caught the flu $t$ weeks after the initial outbreak. . The number of people ill initially is represented by $(0)$ .

STEP 2

To find the number of people who were ill initially, we need to evaluate the function at $t=0$ . $(0)=\frac{5000}{1+999 e^{-0}}$

STEP 3

implify the expression inside the exponential function. Since any number raised to the power of0 is1, we have $e^{-0} = e^{0} =1$ So, the function becomes $(0)=\frac{5000}{1+999 \times1}$

STEP 4

implify the denominator.
$(0)=\frac{5000}{1+999} = \frac{5000}{1000}$

STEP 5

Calculate the value of $(0)$ .
$(0) = \frac{5000}{1000} =5$ So, initially, there were5 people ill. Since we cannot have a fraction of a person, we round to the nearest whole number. Therefore, the answer is5 people.

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