Answer
(a) $C(t) = C_0~e^{-kt}$
(b) It takes 99.7 hours to eliminate 90% of the drug.
Work Step by Step
(a) $C'(t) = -kt$
We know that the solution for this equation is:
$C(t) = C_0~e^{-kt}$
(b) We can find the value of $k$:
$C(t) = C_0~e^{-kt}$
$C(30) = C_0~e^{-30~k} = 0.5~C_0$
$e^{-30~k} = 0.5$
$-30~k = ln(0.5)$
$k = \frac{ln(0.5)}{-30}$
$k = 0.0231049$
We can find the time $t$ it takes until $C(t) = 0.1~C_0$:
$C(t) = C_0e^{-kt}$
$C_0~e^{-kt} = 0.1~C_0$
$e^{-kt} = 0.1$
$-kt = ln(0.1)$
$t = \frac{ln(0.1)}{-k}$
$t = \frac{ln(0.1)}{-0.0231049}$
$t = 99.7~hours$
It takes 99.7 hours to eliminate 90% of the drug.
