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

NameDate

Figure — use these to answer the problems

Two interacting quantities settling to a steady state. Release the system from different starting points and it still ends up in the same place.

how fast it settles1

starting first quantity3

starting second quantity-2

Figure 1

Straightforward practice. Get the method working first.

5 problems

dP/dt = 0.3P(1 − P/450). What is the long-run population?
Answer
2
What should you decide first when building a model?
- What governs the rate of change of the quantity.
- Which solving technique you will use.
A population limited by food. Enter 1 exponential, 2 logistic, 3 cooling.
Answer
A hot object in a cool room. Enter 1 exponential growth, 2 logistic, 3 cooling.
Answer
Predators and prey affecting each other. Enter 1 a single equation, 2 a system.
Answer

The same ideas with more to keep track of.

3 problems

A mass on a spring. What order is the governing equation?
Answer
dP/dt = 0.1P(1 − P/700). What is the long-run population?
Answer
An exponential growth model run for a thousand years. Is it still realistic? 1 yes, 0 no.
Answer

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

4 problems

9
Match each situation to its model.
Draw a line from each item on the left to its match on the right.
- Radioactive decay
- A population with a carrying capacity
- A mass on a damped spring
- Exponential decay
- The logistic equation
- A second-order equation with a velocity term
dT/dt = −k(T − 19). What is the long-run temperature?
Answer
The Choice
A population with unlimited resources. Enter 1 exponential, 2 logistic, 3 cooling.
Answer
The Coffee Again
An object cooling toward a room at 22 degrees. What is the long-run temperature?
Answer