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 real-world 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
- Non-Linear Models
Let’s look at each of these in more detail:
Linear models include a linear variable. We’ll start by looking at models that are basically 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 six-hour period. Another restaurant sold s sandwiches per hour over a seven-hour 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:
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.
Non-linear 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 non-linear model of real-world 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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