F(t)= 30000
1+20e^-1.5t
Describes the number of people, f(t), who have become ill with influenza t weeks after the initial outbreak in a town with 30,000 inhabitants.
a. How many people became ill with flu when the epidemic began?
b. How many people were ill by the end of the fourth week?
Isn't there a sign between 30,000 and 1?
To find the answers to these questions, we need to substitute the given values of 't' into the equation and solve for 'f(t)'.
a. How many people became ill with flu when the epidemic began?
We are looking for the number of people who became ill when t = 0 (the beginning of the epidemic). To find this value, substitute t = 0 into the equation:
F(0) = 30000 / (1 + 20e^(-1.5*0))
Simplifying,
F(0) = 30000 / (1 + 20e^0)
Since any number raised to the power of 0 is 1, the equation becomes:
F(0) = 30000 / (1 + 20)
Simplifying further,
F(0) = 30000 / 21
Calculating,
F(0) ≈ 1428.57
Therefore, approximately 1428 people became ill with the flu when the epidemic began.
b. How many people were ill by the end of the fourth week?
We need to find the number of people who became ill at t = 4 (the end of the fourth week). To find this value, substitute t = 4 into the equation:
F(4) = 30000 / (1 + 20e^(-1.5*4))
Simplifying,
F(4) ≈ 30000 / (1 + 20e^(-6))
Since e^(-6) is a very small value close to 0, it can be approximated as 0 without significant loss of accuracy. Therefore, the equation becomes:
F(4) ≈ 30000 / (1 + 20*0)
F(4) ≈ 30000 / 1
F(4) = 30000
Therefore, at the end of the fourth week, approximately 30,000 people were ill with the flu.
To summarize:
a. Approximately 1428 people became ill with the flu when the epidemic began.
b. Approximately 30,000 people were ill by the end of the fourth week.