There are many different uses for mathematical models such as understanding relationships or making predictions about the future, and there are many different types of mathematical models that can be used depending on the needs of the scientist. Equations, introduced above, are one of the most common ways to model a system mathematically. Graphs are another common method of mathematical modeling, and each of these types of models is explored in detail below.
Equations
Equations are mathematical statements that relate different aspects of a system. They are numerical descriptions that make it possible to understand how the parts of a system work together. For example, recall the kinematic equation from Example 1. This equation stated how initial velocity, acceleration, and time could be used to find final velocity, and, in Example 1, this numerical statement was able to give a quantitate description of how fast the race car and driver were traveling. Example 2 is another example of modeling a system using an equation.
Example 2: A real estate agent has a very picky client who wants a very large basem*nt. This client is so picky that unless the basem*nt has exactly 1750 square feet he absolutely will not buy the house. Use Figure 2 and help the poor agent pick which house has a basem*nt that will work.
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The picky client needs the basem*nt to have a specific area, and the area of a rectangle can be modeled mathematically with the equation {eq}A = lw {/eq} where A stands for area, l stands for length, and w stands for width.
1) Step one is to calculate the area for basem*nt 1. Looking at Figure 2, basem*nt 1 has a length of 25 ft and a width of 55 ft so
{eq}A = 25 * 55 = 1375 {/eq} square feet.
2) Step two is to calculate the area of basem*nt 2 which has a length of 50 ft and a width of 35 ft. The area of this basem*nt is {eq}A = 50 * 35 = 1750 {/eq} square feet. This is the house the client should buy!
Graphs
An equation writes the relationship between quantities, and a graph is a visual representation of that relationship. There are many different types of graphs, and graphs are commonly used mathematical models for data analysis because it is possible to visualize a system without knowing how all of the pieces fit together. Go back to Example 2 and Figure 2. What if the client wanted the basem*nt to have a width of 55 ft and be at least 1750 square feet? How big would the length of the basem*nt have to get so that the area satisfied the client? This problem can be modeled with the equation {eq}A = l*55 {/eq}, and Figure 3 illustrates this problem with the length on the horizontal axis and area on the vertical axis. Following the vertical axis to 1750 then tracing down to the horizontal axis, the basem*nt would need to be at least 31 ft to have the desired square footage.
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Predicting the Future
What does the future hold? How will the system change? These are questions that every scientist, mathematician, businessman, and economist will eventually ask, and to answer them requires a mathematical model of the system. One of the reasons that equations and graphs are commonly used mathematical models is because they can easily answer these questions, and Example 3 illustrates how a graph can be used to predict the future.
Example 3: A small town is quickly growing, and the mayor is worried. Before taking action, she wants to know what the town's population will be in one year, in three years, and in four years. Use Figure 4 to help the mayor find out how fast her town is growing.
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Looking at Figure 3, the population starts out at a little over 100,000 people, and it does not grow much the first year. By year three, the population is almost 300,000. By year four the town's population will be larger than 1,400,000. The poor mayor does indeed have a problem!
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Many branches of math and science use mathematical modeling, and equations and graphs are not the only types of math models. Computer modeling has become a popular method for quantitively understanding real-world problems, and chemical formulas, which succinctly describe chemical reactions, have been around for centuries. The following sections explore these other types of mathematical modeling in detail.
Computer Modeling
Real-world problems often include tens of thousands, if not hundreds of thousands data points, and computer modeling has become the popular method for understanding these large data sets because of the speed and accuracy with which a computer can process information. Computers can take in a large data set, perform statistical analysis on the data, and generate plots within minutes. These statistical analyses and plots model the input data and allow the researcher to understand different attributes of the data set. For example, physicists studying proton-proton collisions at CERN use computer modeling to learn about the particles produced during the collisions. Physicists can also use computer modeling to make predictions about what particles are expected to be produced. This analysis can be done in minutes or hours, but without a computer the same analysis could take decades.
Chemical Formulas as a Math Model
Computer modeling may be new, but chemical formulas as a math model are centuries old. A chemical formula is a mathematical model that relates the relative amounts of composite chemicals in a molecule. For example, the common chemical formula for water, {eq}H_20 {/eq}, means that there are two hydrogen atoms for every one oxygen atom in a molecule of water.
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There are countless math models, and the following list highlights some frequently used mathematical model examples:
- histograms
- scatter plots
- linear equations
- the Pythagorean theorem
- Bayes' formula
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Mathematical modeling is the process of making a numerical or quantitative representation of a system, and there are many different types of mathematical models. The most commonly used math models are equations and graphs. A key feature of equations and graphs that makes them useful models is their ability to make predictions about the future of the system in question. Math models can also be used to learn about a system, visualize data, or understand the origins of the system.
Mathematical modeling is used by scientists, mathematicians, businessmen, and economists. Math models have been around for centuries, and they can describe everything from moving objects to chemical formulas. Recently, computer modeling has become a popular method for understanding large data sets and for running statistical analysis such as using Bayes' formula. Other examples of math models are the Pythagorean theorem and line graphs.
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Video Transcript
Mathematical Models
Suppose you are building a rectangular sandbox for your neighbor's toddler to play in, and you have two options available based on the building materials you have. The sandbox can have a length of 8 feet and a width of 5 feet, or a length of 7 feet and a width of 6 feet, and you want the sandbox to have as large an area as possible. In other words, you want to determine which dimensions will result in the larger area of a rectangle. Thankfully, in mathematics, we have a formula for the area (A) of a rectangle based on its length (l) and width (w).
- A = l × w
Awesome! We can use this formula to figure out which dimensions will make a bigger sandbox!
We can calculate the two areas by plugging in our lengths and widths for each choice:
- A1 = 8 × 5 = 40 square feet
- A2 = 7 × 6 = 42 square feet
We see that a length of 7 feet and a width of 6 feet will result in the larger area of 42 square feet. Problem solved!
Here's something neat! The formula we used to solve this area problem is an example of a mathematical model. A mathematical model is a tool we can use to replicate real-world situations and solve problems or analyze behavior and predict future behavior in real-world scenarios.
Types of Mathematical Models
Let's first take a look at equations.
Equations
The mathematical model we just used was in the form of a formula, or equation. Equations are the most common type of mathematical model.
Here's another example of an equation as a mathematical model. Suppose that a store is having a closeout sale, where everything in the store is 15% off. That is, if an item is x dollars, then the discount is 15% of x, or 0.15x. The sale price can be found by subtracting the discount (0.15x) from the original price (x), giving the following equation that models the sale price of any item in the store with the original price x:
- S = x - 0.15x
We can also combine like terms and write this equation as:
- S = 0.85x
Both of these equations are mathematical models, because they represent a real-world scenario by using a formula to find the sale price of anything in the store. For instance, if something is originally $5, then the sale price can be found using our model by plugging in x = 5:
- S = 0.85(5) = 4.25
We see that the sale price is $4.25.
Graphs
Now, let's take a closer look at graphs.
Another type of mathematical model is a graph. As we just said, most mathematical models are expressed in the form of an equation. Equations can be graphed, so it makes sense that another type of mathematical model would be a graph. For example, we could illustrate the sale prices of store items on a graph, where the y-axis is the sale price, and the x-axis is the original price of an item.
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We can determine the sale price of an item by locating its original price along the x-axis and then finding the corresponding y-value, or sale price, on the graph. As shown on the graph, if an item has an original sale price of $5, then the corresponding sale price is $4.25, which is what we expected based on our findings from the equation. A graph is another tool, or mathematical model, that we can use to understand real-world scenarios.
Other Types of Models
Though equations and graphs are the most common types of mathematical models, there are other types that fall into this category. Some of these include pie charts, tables, line graphs, chemical formulas, or diagrams.
Let's take a closer look at these now.
Chemical Formulas
A chemical formula is an expression that specifies the types and numbers of atoms present in a molecule. It can take on the following form:
- (Atom abbreviation)N, where N is the number of the atoms in the molecule.
A common example of a chemical formula would be the one for water, or H2O. This is because the oxygen atom has no subscript, there is only one of that type of atom. Therefore, this formula indicates that one molecule of water contains two hydrogen atoms (H2) and one oxygen atom (O). Some other common chemical formulas are shown in the following table:
| Chemical Name | Common Name | Number and Type of Atom |
|---|---|---|
| NaCl | Table salt | 1 sodium (Na), 1 chlorine (Cl) |
| C2 H6 O | Alcohol | 2 carbon (C2), 6 hydrogen (H6), 1 oxygen (O) |
| NH3 | Ammonia | 1 nitrogen (N), 3 hydrogen (H3) |
| CO2 | Carbon dioxide | 1 carbon (C), 2 oxygen (O2) |
A chemical formula is a good example of a mathematical model that might have escaped consideration. The formulas themselves don't look like what we're used to seeing in mathematics, but they are still a mathematical way of modeling molecules.
Lesson Summary
Let's take a couple moments to review.
Mathematical models are tools we can use to approach real-world situations mathematically. While there are many types of mathematical models, the most common one is the equation. We can model a given scenario with an equation, or formula, and use that equation to answer different questions about the scenario. Another common mathematical model is a graph, which can be used to model different scenarios in the same way we use equations.
Some lesser-known mathematical models, but still equally important, are pie charts, diagrams, line graphs, chemical formulas, or tables, just to name a few. We can use a chemical formula to specify the types and numbers of atoms present in molecules we find in our everyday lives, such as good old H2O.
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