Mathematical Models

Mathematics can be used to "model", or represent, how the real world works.

So we have a (very simple) mathematical model of the space in that box.

Accurate?

The model is not the same as the real thing.

In our example we did not think about the thickness of the cardboard, or many other "real world" things.

But hopefully it is good enough to be useful.

If we are charged by the volume of the box we send, we can take a few measurements and know how much to pay. It can also be useful when deciding which box to buy when we need to pack things. So the model is useful!

But maybe we need more accuracy, we might need to send hundreds of boxes every day, and the thickness of the cardboard matters. So let's see if we can improve the model:

Now we have a better model. Still not perfect (did we consider wasted space because we could not pack things neatly, etc ...), but better.

So a model is not reality, but should be good enough to be useful.

Playing With The Model

Now we have a model, we can use it in different ways:

So the model is useful. It lets us know we will get 2% more space inside the box (for the same outside measurements).

But there are still "real world" things to think about, such as "will it be strong enough?"

Thinking Clearly

To set up a mathematical model we also need to think clearly about the facts!

Here is another one:

Example: You are the supervisor of 8-hour shift workers. They recently had their break times reduced by 10 minutes but total production did not improve.

At first glance there is nothing to model, because there was no change in production.

But wait a minute ... they are working 10 minutes more, but producing the same amount, so production per hour must have dropped!

Let us assume they used to work 7 hours (420 minutes):

Change in production per hour = 410/420 = 0.976...

Which is a reduction of more than 2%

But even worse: the first few hours of the shift are not affected by the shorter break time, so it could be a 4 or 5% reduction later in the shift.

You could recommend:

• looking at production rates for every hour of the shift
• trying different break times to see how they affect production

A Bigger Example: Most Economical Size

OK, let us have a go at building and using a mathematical model to solve a real world question.

Step One: Draw a sketch!

It helps to sketch out what we are trying to solve!

The base is square, so we will just use "w" for both lengths

The box has 4 sides, and double tops and bottoms. The box shape could be cut out like this (but is probably more complicated):

Step Two: Make Formulas.

Ignoring thickness for this model:

Volume = w × w × h = w²h

And we are told that the volume should be 0.02m³:

w²h = 0.02

Areas:

Step Three: Make a Single Formula For Cost

We want a single formula for cost:

Cost = $0.30 × Area of Cardboard

Cost = $0.30 × (4wh + 4w²)

And that is the cost when we know width and height.

That could be hard to work with ... a function with two variables.

But we can make it simpler! Because width and height are already related by the volume:

Volume = w²h = 0.02

... which can be rearranged to ...

h = 0.02/w²

... and that can be put into the cost formula ...

Cost = $0.30 × (4w×0.02/w² + 4w²)

And now the cost is related directly to width only.

With a little simplification we get:

Step Four: Plot it and find minimum cost

What to plot? Well, the formula only makes sense for widths greater than zero, and it turns out that for widths above 0.5 the cost continues to rise.

So here is a plot of that cost formula for widths between 0.0 m and 0.55 m:

Plot of y= 0.3(0.08/x+4x²)
x is width, and y is cost

Just by eye, we see the cost reaches a minimum at about (0.22, 0.17). In other words:

• when the width is about 0.22 m (x-value),
• the minimum cost is about $0.17 per box (y-value).

In fact, looking at the graph, the width could be anywhere between 0.20 and 0.24 without affecting the minimum cost very much.

Step Five: Recommendations

Using this mathematical model we can now recommend:

• Width = 0.22 m
• Height = 0.02/w² = 0.02/0.22² = 0.413 m
• Cost = $0.30 × (0.08/w+ 4w²) = $0.30 × (0.08/0.22+ 4×0.22²) = $0.167

Or about 16.7 cents per box

But any width between 0.20 m and 0.24 m is fine.

We can also suggest improvements to this model:

• Include cost of glue/staples and assembly
• Include wastage when cutting box shape from cardboard.
• Is this box a good shape for packing, handling and storing?
• Any other ideas?

Predicting the Future

Mathematical models can also be used to forecast future behavior.

Computer Modeling

Mathematical models can get very complex, and so the mathematical rules are often written into computer programs, to make a computer model.

Have a play with a simple computer model of reflection inside an ellipse
or the single pendulum or double pendulum animation.

More complex examples include:

• Weather prediction
• Economic Models (predicting interest rates, unemployment, etc)
• Public health vs infectious diseases
• Models of how large structures behave under stress (bridges, skyscrapers, etc)
• Many more ...

If you become an expert in any of those you will have a job for life!