 # Using Mathematical Models to Solve Problems

Mathematical modelling simply refers to the creation of mathematical formulas to represent a real world problem in mathematical terms. This lesson reviews the creation and pitfalls of mathematical models.

## Introduction to Modelling

Hi and welcome to this lesson on mathematical modelling. In order to understand mathematical modelling, it is first important to understand the term modelling. In the most broad definition of the word, modelling could be said to be showing an example of a scenario.

When a model is on the runway, he or she is showing an example of what the clothing would look like on a person. Many people enjoy building model cars or planes. Again, these models give an example of what real cars and airplanes look like. Mathematical modelling is the same – it simply refers to the creation of mathematical formulas to represent a real-world problem in mathematical terms. Join me, now, as we look closer at the use of mathematical modelling in real-world situations.

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## Mathematical Model

You’re probably already aware of some very well-known mathematical models – those that give perimeter and area of a square. The models are simply the formulas P = 2(L) + 2(W) and A = L * W (P is perimeter, A is area, L means length and W means width). So, yes, these formulas are mathematical models because they are examples or representations of the image that can be used repeatedly with different values to obtain different results for the same scenario. Mathematical models can be used to model real-world scenarios as well as pure math scenarios.

If you were in charge of purchasing the fruit for your book group each week, how could you go about figuring out how much money was needed each week for your purchase? Well, one way would be to work it out from scratch each week. Or, you could build a mathematical model to assist you in getting your total faster each week.

Let’s assume you have been doing this enough to know that you will need 3 apples, 2 oranges, and 4 bananas each time. But, the price of fruit changes slightly from week-to-week, so you can’t just assume from one week to the next how much it will cost. You know that you will need to multiply the price of apples times 3, the price of oranges time 2, and the price of bananas by 4 each week.

This is the perfect start to a mathematical model. Here is a formula you might write to model this scenario: 3(a) + 2(o) + 4(b) = your total. Notice that I used the first initial of each fruit to represent the price. In math, you don’t have to use x and y for variables, any letter can act as an unknown.

So, in our example, if we find one week that apples are \$0.35 each and oranges are \$0.25 each and bananas are \$0.50 each, our model would work out to 3(.35) + 2(.25) + 4(.5) = \$3.55.

If, the next week, oranges go up to \$0.75 each, then we just replace this value in our model to work out the new cost. This mathematical model will work each week as we calculate the total cost of fruit for the group.

## Not Always Accurate

One thing to remember about mathematical models is that they are not always accurate. By that I mean that they are not necessarily precise in real-world scenarios. Take the example of the volume of a box. We know a simple mathematical model for determining the volume of a box is L * W * H, right?

This works fine in theory, but in practice (that is to say, in the real world actually doing the math) there is something missing. The original model does not take into account the actual cardboard itself. Depending on how thick or thin it is, the outer measurements are not the same as the inner measurements; for shipping purposes, the outer measurements of the box are noted, but for packing purposes, the inner measurements of the box are more important for volume.

A more accurate model for the volume of the box (for packing purposes) would have to take into account the thickness of the cardboard itself: if t is thickness, then volume = (L – 2t) * (W – 2t) * (H – 2t).

Similarly, in our fruit purchasing model, the model does not accurately take into account all of the possible expenses incurred to purchase fruit. A simple exclusion is the fuel required to travel to and from the fruit market. Another could be the expense for parking (if there are parking meters at the market).

So, you see, while a model is a close representation of the real world, it is not completely accurate. The more variables you can think of to add to your model, the more accurate the model will be.

## Lesson Summary

We have seen that a mathematical model is just a representation of a real-world scenario in formula form. It follows from any model, which is an example scenario. To make a mathematical model, all you need to do is devise a formula to represent the variables in your scenario. This gives you one model to use in every instance of a similar scenario (such as our fruit purchase example).

Just as with more common physical models, mathematical models are not perfectly accurate. They are representations of a perfect scenario, but we all know the real world is not perfect. Just because a shirt looks great on the model, does not necessarily mean it will look good on me in the real world. To combat this accuracy issue with modelling, it is important to add as many variables into your model as possible. The more variables you have accounted for, the more accurate your model will be. Thanks for joining me. Bye.

## Learning Outcomes

You’ll have the ability to do the following after this lesson:

• Define mathematical model
• Explain how to make a mathematical model
• Describe the limitations of mathematical models
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