A large tank holds of pure water

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Mathematical Methods for Physics, Ch First-order differential equations and physical modelling

Problem

A large tank holds

100\,\mathrm{L}

of pure water. From

t=0

, brine carrying

20\,\mathrm{g}

of salt per litre flows in at

3.0\,\mathrm{L\,min^{-1}}

, the tank is stirred so that its contents stay uniform, and the mixture drains out at

2.0\,\mathrm{L\,min^{-1}}

. The tank can hold

500\,\mathrm{L}

. Let

m(t)

be the mass of salt in the tank, with

t

in minutes. ((a)) Write a differential equation for

m(t)

, and state its order and whether it is linear. ((b)) Solve it with

m(0)=0

, and find the salt concentration in the tank at

t=100\,\mathrm{min}

. ((c)) Find when the concentration first reaches

19.0\,\mathrm{g\,L^{-1}}

. ((d)) The concentration creeps up towards the

20\,\mathrm{g\,L^{-1}}

of the incoming brine. Find how long the gap between the two takes to halve, first starting from

t=0

and then starting from

t=100\,\mathrm{min}

. Decide whether this approach has a single time constant, as the exponential approaches did.

Related theory

Ordinary differential equations

Browse the problem library