Question

Under certain conditions, orange is converted into juice at a rate, which is proportional to the

amount unconverted at any time. If out of 70 grams of juice at t = 0, 6 grams are converted

during the first 3 minutes, find the amount converted in 1.5 hours.

Answer 1

$x-remaining\,\,volume\,\,of\,\,juice\\x_0-starting\,\,volume\\\frac{dx}{dt}=-kx\Rightarrow x=x\left( 0 \right) e^{-kt}=x\left( 0 \right) \lambda ^t\\x\left( 0 \right) =70,x\left( 3 \right) =70-6\Rightarrow 64=70\lambda ^3\Rightarrow \lambda =\sqrt[3]{\frac{64}{70}}\\1.5hours=90\min \\x\left( 90 \right) =x\left( 0 \right) \lambda ^{90}=70\cdot \left( \sqrt[3]{\frac{64}{70}} \right) ^{90}=4.75944-remaining\\x\left( 0 \right) -x\left( 90 \right) =70-4.75944=65.2406-converted$