Problem 10 - Algebra 1: Concepts and Skills

Question

Write an exponential function to model the situation. Tell what each variable represents. Your salary of $25,000 increases 7% each year.

Step-by-Step Solution

Answer

The exponential function modelling the scenario is \(F(t) = 25000*(1 + 0.07)^t\), where \(F(t)\) is the salary after \(t\) years, \(t\) is the number of years, \(25000\) is the starting salary, and \(0.07\) is the annual rate of increase.

Step 1: Identify Variables

The initial salary is $25,000, denoted as \(P\). The rate of increase (as a decimal) is 7% or 0.07, denoted as \(r\). The variable representing the number of years is \(t\).

Step 2: Formulate the Exponential Function

Exponential functions to model a situation like this are typically in the form \(P*(1 + r)^t\). Here, plug in the values: \(P = 25000\), \(r = 0.07\). So the function becomes \(F(t) = 25000*(1 + 0.07)^t\).

Step 3: Interpret the Variables

In \(F(t) = 25000*(1 + 0.07)^t\), \(F(t)\) represents the salary after \(t\) years, \(t\) is the number of years, \(25000\) is the initial salary, and \(0.07\) is the rate at which the salary increases every year.

Key Concepts

Percent Increase

Understanding percent increase is crucial when determining how a value grows over time. In the context of salary, it means how much more you make compared to the previous year.
For our example, your salary increases by 7% each year. Here's how it works:

• The percent increase is expressed as a fraction of 100. So, 7% becomes \(0.07\) in decimal form. This is the amount your salary grows each year relative to the previous year's salary.
• If your salary was initially \(25,000\), a 7% increase would add \(25,000 \times 0.07 \), or \(1,750\), more to your salary next year.

By understanding the concept of percent increase, you can predict how much your salary will be for each future year.

Variable Identification

Identifying variables is a fundamental step in setting up mathematical models, especially for exponential functions.
It's essential to assign what each letter in your equation represents to avoid confusion.

• In our exercise, we have several important variables:
  • \(P\): This denotes the initial value. Here, it's your starting salary, \(25,000\).
  • \(r\): The rate of increase, expressed as a decimal. In our example, it's \(0.07\), which corresponds to a 7% increase.
  • \(t\): The time in years over which the salary is increasing. It's a variable that changes depending on how many years into the future you're calculating for.

Clear variable identification helps ensure that the math you've laid out represents the situation accurately.

Mathematical Modeling

Mathematical modeling is a method used to represent real-life scenarios through mathematical expressions.
For salary increases, an exponential function models the situation well, as it describes continuous growth over time.

• The basic format for an exponential growth model is \(F(t) = P(1 + r)^t\). This equation allows you to calculate future values based on current data and a rate of change.
• The expression \((1 + r)\) signifies how each year's salary builds on top of the prior year's results. The \(t\) exponent denotes repeated multiplication over time.
• By substituting \(P = 25,000\) and \(r = 0.07\), our function becomes \(F(t) = 25,000(1 + 0.07)^t\). This expression can be used to predict salary growth over any number of years you choose.

Mathematical models like this make it easier to calculate future outcomes and understand how various factors interact to influence those outcomes.