Question

A cup of coffee contains about 100 mg of caffeine. Caffeine is metabolized and leaves the body at a continuous rate of about 12% every hour. a. Write a differential equation for the amount, A, of the caffeine in the body as a function of the number of hours, t, since the coffee was consumed. b. Use the differential equation to find dA/dt at the start of the first hour (right after the coffee is consumed). Use your answer to estimate the change in the amount of caffeine during the first hour.

Answer 1

The differential equation for the caffeine amount in the body is dA/dt = -0.12A. Right after consumption, caffeine is metabolized at a rate of 12 mg/hr, so about 12 mg is lost in the first hour.

Step-by-step explanation:

Creating the Differential Equation

To find the differential equation for the amount of caffeine in the body, we need to consider the rate of metabolism of caffeine. If the initial amount of caffeine is 100 mg and the rate at which it leaves the body is 12% per hour, let A represent the amount of caffeine in the body at time t, where t is the number of hours since the coffee was consumed. The rate of change of the amount of caffeine can be represented by the differential equation dA/dt = -0.12A, where the negative sign shows that the amount of caffeine is decreasing.

Estimating Caffeine Change During the First Hour

To find dA/dt at the start of the first hour, substitute A = 100 mg into the equation, obtaining dA/dt = -0.12(100) = -12 mg/hr. This means that at the start of the first hour, the caffeine is metabolized at a rate of 12 mg per hour. Using this rate, we can estimate that the change in the amount of caffeine during the first hour is approximately 12 mg.

Answer 2

a.
`A = C_(0)(1-x)^t\\x: percentage\ of \ caffeine\ metabolized\\`

b.
`(dA)/(dt)= -11.25 (mg)/(h)`

Step-by-step explanation:

First, we need tot find a general expression for the amount of caffeine remaining in the body after certain time. As the problem states that every hour x percent of caffeine leaves the body, we must substract that percentage from the initial quantity of caffeine, by each hour passing. That expression would be:

`A= C_(0)(1-x)^t\\t: time \ in \ hours\\x: percentage \ of \ caffeine\ metabolized\\`

Then, to find the amount of caffeine metabolized per hour, we need to differentiate the previous equation. Following the differentiation rules we get:

`(dA)/(dt) =C_(0)(1-x)^t \ln (1-x)\\(dA)/(dt) =100*0.88\ln(0.88)\\(dA)/(dt) =-11.25 (mg)/(h)`

The rate is negative as it represents the amount of caffeine leaving the body at certain time.