Question

Solve as a difference equation.

Answer 1

Solution :

Let y = amount of caffeine at a given time in body.

n = number of hours

So $\frac{dy}{dn}= - \ Y$

$\Rightarrow \frac{dY}{Y}=- 0.01 \ dn$

ln Y = -0.01 n + C

At n = 0, Y = 130 mg

ln 130 = -0.01 x 0 + C

C = ln 130

$\ln \frac{Y}{130} =- 0.01 \ n$

$Y = 130 \ e^{-0.01n}$

When Y = 65

$65 = 130 \ e^{-0.01 n }$

$\frac{65}{130}=e^{-0.01n}$

$\ln \frac{1}{2} = -0.01 \ n$

n = 69.315 hours

At n = 24 hours

$Y = 130 \ e^{-0.01 \times 24}$

Y = 102.26 mg