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| Math Central | Quandaries & Queries |
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The function f(t) = a/(1+3e^(-bt)) has also been used to model the spread of a rumor. Suppose that a=70 and b=0.2. Compute f(2), the percentage of the population that has heard the rumor after 2 hours. Compute f'(2) and describe what it represents. Compute lim t approaches infinity and describe what it represents. Can I please get help with this calculus problem? |
Hi,
The function that is used to model this rumor is
\[f(t) = \frac{70}{1 + 3 e^{-0.2 t}}\]
where $t$ is the number of hours after the rumor was started and $f(t)$ is the percentage of people who have heard the rumor at time $t.$ Substitute $t = 2$ and use your calculator to evaluate $f(2).$ I got a little over $23 \%.$
Use the calculus you know to obtain $f'(t)$ and then calculate $f'(2).$ This will tell the rate at which the rumor is spreading at time $t,$ in percentage per hour.
The last sentence asks you to evaluate the limit of $f(t)$ as $t$ approaches infinity. You know that as $t$ approaches infinity $e{-t}$ approaches zero, so this limit is straightforward to evaluate. This limit is a percentage. What does it represent?
Penny
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