Modelling the Real World

Everything this course was building toward.

12 problems · about 24 minutes · A-CED.A.1, A-CED.A.2, F-LE.A.1

What this lesson teaches

The student selects an appropriate model for a described situation and interprets its behaviour.

Choosing a model starts from what governs the rate of change, not from what you know how to solve.
Interacting quantities need a system; forces need a second-order equation.
Equilibria and stability usually answer the real question, and every model has a working range.

Straightforward practice. Get the method working first.

5 problems

dP/dt = 0.3P(1 − P/450). What is the long-run population?
Answer 450
Why 450.
What should you decide first when building a model?
Answer What governs the rate of change of the quantity.
Why What drives the rate.
A population limited by food. Enter 1 exponential, 2 logistic, 3 cooling.
Answer 2
Why There is a carrying capacity.
A hot object in a cool room. Enter 1 exponential growth, 2 logistic, 3 cooling.
Answer 3
Why The rate depends on a temperature gap.
Predators and prey affecting each other. Enter 1 a single equation, 2 a system.
Answer 2
Why Two interacting quantities.

The same ideas with more to keep track of.

3 problems

A mass on a spring. What order is the governing equation?
Answer 2
Why Force relates to acceleration.
dP/dt = 0.1P(1 − P/700). What is the long-run population?
Answer 700
Why The carrying capacity.
An exponential growth model run for a thousand years. Is it still realistic? 1 yes, 0 no.
Answer 0
Why Every model has a working range.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each situation to its model.
Answer Radioactive decay → Exponential decay; A population with a carrying capacity → The logistic equation; A mass on a damped spring → A second-order equation with a velocity term
Why Only one of the three involves a force.
dT/dt = −k(T − 19). What is the long-run temperature?
Answer 19
Why The stable equilibrium.
The Choice: A population with unlimited resources. Enter 1 exponential, 2 logistic, 3 cooling.
Answer 1
Why Exponential.
The Coffee Again: An object cooling toward a room at 22 degrees. What is the long-run temperature?
Answer 22
Why 22 degrees.