# Mr Job's Maths Class

## Lessons

### Non-linear models

Topics:

From 1935-37, the American marine toad (Bufo marinus) was introduced into Queensland, Australia in eight coastal sugar cane districts. Due to lack of natural predators and an abundant food supply, the population grew and the poisonous toads began to be found far from the region in which they were originally introduced.

The data* in the table below shows how far “cane toads” spread in the subsequent years.

Year Area occupied (km2)
1939 32 800
1944 55 800
1949 73 600
1954 138 000
1959 202 000
1964 257 000
1969 301 000
1974 584 000

*Source: http://www.geom.uiuc.edu/education/calc-init/population/unbounded.html

• This data can be entered into Desmos by changing Year to time in years since 1939. That is 1939 is zero, 1944 is 5.
The area occupied can be entered in units of thousands (e.g. 32.8, 55.8 …).
• This data takes roughly an exponential form.
Create a model in Desmos to fit the data (approximately).
Use the model y = 32.8(a)x where a is a slider with value 1 ⩽ a ⩽ 1.2 with steps of 0.01.
Fit the model to the data.
1. What is the y-intercept? What does this value represent?
2. From the model, predict the area occupied by cane toads in 1989.
3. The year 2017 is 78 years since 1939.
1. From the model, predict the area occupied by cane toads in 2017.
2. The area of Australia, excluding the Australian Antarctic Territory, is 7.692 million km². How does this compare to your prediction in a.? From this, what could you say about the model you’ve developed?

#### Modelling compound interest

Consider an exponential model studied elsewhere in this course (later on).

Future value of a compound interest investment (MS-F4 F4.1).

Given FV = PV(1 + r)n. This is of the form y = b(a)x.
For example, for an investment of \$1000 invested at 5% p.a. compounded annually.

FV = 1000(1 + 0.05)n

this can be simplified to:

FV = 1000(1.05)n

Use Desmos to graph and investigate.
Scenario: You are borrowing \$2000 to \$20000 to purchase your first car. Use Desmos to model the compound interest for rates of 1% p.a. to 10% p.a.

What can you say about the y-intercept of this exponential graph?

See the video below for help.

Wolfram Alpha can do many maths questions.

Scientific Calculator