A cup of coffee contains about 100 𝑚𝑔 of caffeine. Caffeine is metabolized and leaves the bodyat a continuous rate of about 17% every hour. Write a differential equation for the amount, 𝐴,of caffeine in the body as a function of the number of hours, 𝑡, since the coffee was consumed
Question
A cup of coffee contains about 100 𝑚𝑔 of caffeine. Caffeine is metabolized and leaves the bodyat a continuous rate of about 17% every hour. Write a differential equation for the amount, 𝐴,of caffeine in the body as a function of the number of hours, 𝑡, since the coffee was consumed
Solution
The amount of caffeine in the body decreases by 17% every hour. This can be represented by a first order linear differential equation.
Let A(t) be the amount of caffeine in the body at time t. The rate of change of A with respect to time, dA/dt, is proportional to the current amount of caffeine, A(t). The constant of proportionality is -0.17 (since the amount of caffeine decreases by 17% every hour).
So, the differential equation is:
dA/dt = -0.17 * A(t)
This equation says that the rate of change of the amount of caffeine in the body is -17% of the current amount of caffeine.
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