SME
Differential Equations · Problem type
A mixing tank
Use this for “A 400 L tank of fresh water takes in brine with 25 g/L of salt at 8 L/min, and the mixture drains at the same rate. How much salt is in it after 30 minutes?”
Chart
Horizontal asymptote A = 2,000
A(20) ≈ 1,411
This is an example. Type your own values to start a new problem.
dA
dt
=
r (L/min)
·
cᵢₙ (g/L)
−
r
V
A
A(0)
=
A₀ (g)
A(
t (min)
)
=
A (g)
A∞
Salt in the long run
worked out (not typed) · g
c(t)
Concentration in the tank at t
worked out (not typed) · g/L
Formulas
A∞
=
V
×
cᵢₙ
2,000 = 100 × 20
A
=
A∞
+
(
A₀
−
A∞)
×
e
−
r
×
t
÷
V
1,411.3929
=
2,000
+
(400
−
2,000)
×
e
−5 × 20 ÷ 100
c(
t
)
=
A
V
14.1139
=
1,411.3929
100
Assumptions
•
The tank is stirred so it is the same all through, and liquid drains at the rate r it flows in, so the volume V stays the same.
•
Salt comes in at r·cᵢₙ and leaves at r·(A ÷ V): dA/dt = r·cᵢₙ − (r ÷ V)A, a linear equation that is also separable.
•
Its solution is A = V·cᵢₙ + (A₀ − V·cᵢₙ)e−rt/V. A moves from A₀ toward V·cᵢₙ and never passes it.
•
Volume is in L, flow in L/min, concentration in g/L, salt in g and time in min.
Step-by-step
WE KNOW
V = 100 L, r = 5 L/min, cᵢₙ = 20 g/L, A₀ = 400 g, t = 20 min
FIND
salt in the long run (A∞), salt at time t (A), concentration in the tank at t (c(t))
Step 1 · Find salt in the long run (A∞)
Use A∞ = V × cᵢₙ
In
the
long
run
the
whole
tank
has
the
inflow’s
concentration,
so
A∞
=
V
·cᵢₙ.
A∞ = 100 × 20
A∞ = 2,000 g
Step 2 · Find salt at time t (A)
Use A = A∞ + (A₀ − A∞) × e−r × t ÷ V
The
gap
A₀
−
A∞
shrinks
by
the
factor
e
−
rt
/
V
.
Add
what
is
left
of
it
to
A∞.
dA/dt
=
5
×
20
−
(
5
100
)
A
A
(
t
)
=
2,000
−
1,600
e
−0.05
t
A
(20)
=
2,000
−
1,600e⁻¹
=
2,000
−
1,600
×
0.367879
=
1,411.3929
A
=
1,411.393
g
Step 3 · Find concentration in the tank at t (c(t))
Use c(t) = A ÷ V
The salt spread through the tank’s volume.
c(
t
)
=
1,411.393
100
c(
t
)
=
14.1139
g/L
Check
Put the numbers back in. Both sides should match.
2,000 = 100 × 20 ✓
1,411.393
=
2,000
+
(400
−
2,000)
×
e
−5 × 20 ÷ 100
✓
14.1139
=
1,411.393
100
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
First-order ODEs
Main lesson
Logistic growth
Use this for “A pond can hold 2000 fish. It is stocked with 200 and grows by dP/dt = 0.3P(1 − P/2000), t in years. How many fish are there after 5 years?”
Euler’s method
Use this for “Use Euler’s method with h = 0.25 to estimate y(1) for y′ = 2x − y, y(0) = 3. How far is it from the exact value?”
A linear equation by an integrating factor
Use this for “Solve y′ + 3y = 6x + 1 with y(0) = 2, and find y(0.5).”
Separable, linear, both or neither
Use this for “Is y′ = y cos x separable, linear, or both? Which method solves it?”