Question

A typical cup of coffee contains about 100 mg of caffeine and every hour approximately 16% ofthe amount of caffeine inthe body is metabolized and eliminated.a)Let C represent the amount of caffeine in the body in mg and f represent the number of hours since a cup of coffee was consumed. Write C as a function of r.
b) How much caf[eine is in the body after 5 hours?

Answer 1

Answer:

a) $\frac{dC}{dt} = rC$

And for this case we can rewrite the model like this:

$\frac{dC}{C} = r dt$

If we integrate both sides we got:

$ln C = rt + k$

If we use exponentials for both sides we got:

$C = e^{rt} e^k = C_o e^{rt}$

For this case $C_o = 100 mg$ and r = -0.16

So then our model would be given by:

$C(t) = 100 e^{-0.16 t}$

Where t represent the number of hours

b) $C(5) = 100 e^{-0.16*5}= 44.9329$

Step-by-step explanation:

Part a

For this case we can assume the proportional model given by:

$\frac{dC}{dt} = rC$

And for this case we can rewrite the model like this:

$\frac{dC}{C} = r dt$

If we integrate both sides we got:

$ln C = rt + k$

If we use exponentials for both sides we got:

$C = e^{rt} e^k = C_o e^{rt}$

For this case $C_o = 100 mg$ and r = -0.16

So then our model would be given by:

$C(t) = 100 e^{-0.16 t}$

Where t represent the number of hours

Part b

For this case we can replace the value t=5 into the model and we got:

$C(5) = 100 e^{-0.16*5}= 44.9329$