Mathematical Models
Overview
Expect to see a fairly large number of questions on the SAT that deal with mathematical models. Basically, a mathematical model is an equation, formula, or expression that can be used to solve a realworld phenomena or problem. The types of questions you’ll encounter will have you to interpret a mathematical model or develop one based on information provided.
In this section, we will review the following aspects of this topic:
 Linear Models
 NonLinear Models
Let’s look at each of these in more detail:
Linear Models
Linear models include a linear variable. We’ll start by looking at models that are basically linear expressions
Linear Expressions
Review the topic of Linear Expressions. Here the goal is to have you build a mathematical model based on a verbal description. Here is an example.
One restaurant sold p pizzas every hour over a sixhour period. Another restaurant sold s sandwiches per hour over a sevenhour period. Write the expression that represents the total number of pizzas and sandwiches sold.
Both the variables p and s are rates. To find the total number of pizzas and sandwiches sold, multiply each rate by the number of hours:

Linear Equations
Look for questions that ask you to analyze or build a mathematical model using a linear equation. Let’s look at an example.
A restaurant sold p pizzas in a day, and each pizza cost $10. It also sold d drinks and each drink was $2.
On a given day 175 pizzas and sodas were sold and generated $750 in revenue. How many pizzas were sold?
Here is the mathematical model for finding the dollar amount of pizzas and sodas sold:
Here is the model for finding the total number of pizzas and drinks sold:
This is a system of equations, which can be solved by the Substitution Method:
So, 50 pizzas were sold.

NonLinear Models
Nonlinear models can include quadratic equations or polynomial equations. You should review the Quadratic Functions and Equations section. Expect to see questions that involve either analyzing or developing a nonlinear model of realworld situation. Let’s look at an example.
A ball is kicked from the ground level. The height (h) of the ball above the ground is modeled by this equation:
In the equation the variable t is the elapsed time (in seconds). The ball is kicked with a vertical speed of 20 meters per second. How long is the ball in the before it lands on the ground?
The equation shown is an example of a quadratic model. To find when the ball hits the ground is the same as solving this quadratic equation:
This quadratic is easily factored to find the two solutions:
The ball will hit the ground after 4 seconds.

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