# 3.2 compartmental analysis 3.2 compartmental analysis mixture problems in these mixture problems a...

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3.2 Compartmental Analysis

Mixture Problems

In these mixture problems a substance dissolved in a liquid is entering and exiting a holding tank. The

differential equation modeling this situation is:

π ππ‘π ππ πΆβππππ ππ π₯(π‘) = π ππ‘π π₯(π‘) πππ‘πππ β π ππ‘π π₯(π‘) ππ₯ππ‘π

ππ₯

ππ‘ = πππ β πππ’π‘

where,

π₯(π‘) = ππππ’ππ‘ ππ π π π’ππ π‘ππππ πππ π πππ£ππ ππ π ππππ’ππ ππ‘ π‘πππ π‘

πππ = (ππππ€ πππ‘π ππ ππππ’ππ πππ‘πππππ) Γ (πππππππ‘πππ‘πππ ππ π π’ππ π‘ππππ ππ ππππ’ππ πππ‘πππππ)

πππ’π‘ = (ππππ€ πππ‘π ππ ππππ’ππ ππ₯ππ‘πππ) Γ (πππππππ‘πππ‘πππ ππ π π’ππ π‘ππππ ππ ππππ’ππ ππ₯ππ‘πππ)

Examples

1. Water flows into a pond at a rate of 10 π3 πππβ and out at the same rate. The volume of the pond is 1000 π3.

Initially, the pond contains 100 grams of pollutants. The concentration of pollutants in the water flowing into the

pond is 2 π π3β . Assuming immediate and uniform dispersion of the pollutants, find a formula for the number

of grams of pollutants in the pond, π₯ = π₯(π‘).

πΆππππππ‘πππ‘πππ ππ π π’ππ π‘ππππ ππ ππππ’ππ ππ₯ππ‘πππ = ππππ’ππ‘ ππ π π’ππ π‘ππππ ππ π‘βπ π‘πππ ππ‘ πππ¦ π‘πππ, π‘

π£πππ’ππ ππ π€ππ‘ππ ππ π‘βπ π‘πππ ππ‘ πππ¦ π‘πππ, π‘

2. The air in a small room 12 ft by 8 ft by 8 ft is 3% carbon monoxide. Starting at π‘ = 0, fresh air containing no

carbon monoxide is blown into the room at a rate of 100 ππ‘3 πππβ . If air in the room flows out through a vent

at the same rate, when will the air in the room be 0.01% carbon monoxide?

Population Models

ππ

ππ‘ = ππ, π(0) = π0

Where k is a constant of proportionality

Note: π > 0 for growth and π < 0 for decay

3. Solve the differential equation for population growth.

ππ

ππ‘ = ππ, π(0) = π0

This is the Malthusian or exponential, law of population growth and it assumes that members of a population only die

by natural causes. This means it is not very accurate for a large population or over a long period of time.

The Logistic Equation

The logistic equation has proven to be more accurate in predicting population growth because it takes into account

competition within a population.

ππ

ππ‘ = βπ΄π(π β π1) , π(0) = π0

where a and b are constants.

Solving this equation gives us the logistic function

π(π‘) = π0π1

π0 + (π1 β π0)π βπ΄π1π‘

4. In 1980 the population of alligators on the Kennedy Space Center grounds was estimated to be 1500. In 2006

the population had grown to an estimated 6000.

a. Use the Malthusian law for population growth to estimate the alligator population on the Kennedy

Space Center grounds in the year 2020.

b. Suppose we have the additional information that the population of alligators on the grounds of the

Kennedy Space Center grounds in 1993 was estimated to be 4100. Use the logistic model to estimate the

population of alligators in the year 2020. What is the predicted limiting population?

π1 = [ ππππ β 2π0ππ + π0ππ

ππ 2 β π0ππ

] ππ

π΄ = 1

π1π‘π ln [

ππ(ππ β π0)

π0(ππ β ππ) ]

5. If initially there are 300g of a radioactive substance and after 5 years there are 200g remaining, how much time

must elapse before only 10g remain?

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