For a mixing situation in which liquid enters and leaves at equal rates so that the…
MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations
Problem
For a mixing situation in which liquid enters and leaves at equal rates so that the tank volume V remains constant, which of the following best explains why the differential equation governing the quantity of dissolved substance Q(t) can be solved by direct separation of variables?
(a) Because equal flow rates cancel the inflow and outflow terms, leaving dtdQ=0, so separation is unnecessary.
(b) Because V is constant, the outflow term is proportional only to Q, giving an equation of the form dtdQ=a−bQ, whose variables can be separated.
(c) Because the inflow concentration is always constant, the equation is linear but never separable into Q and t terms.
(d) Because the outflow term can be written as (Q/V)r, the inflow term disappears, giving dtdQ=−VQ, which does not allow full separation of Q and t."