Problem 57 - Algebra 1

Question

From 1995 through 1999 , the sales for a national furniture store increased by about the same percent each year. The sales \(s\) (in millions of dollars) for year \(t\) can be modeled by \(s=476(1.13)^{t},\) where \(t=0\) represents \(1995 .\) Find the ratio of 1997 sales to 1999 sales.

Step-by-Step Solution

Answer

The ratio of the sales in 1997 to the sales in 1999 is 0.81 or 81%.

Step 1: Identify and Substitute

Firstly, identify the values for \(t\) for the years 1997 and 1999. Since \(t=0\) represents the year 1995, \(t=2\) represents 1997 and \(t=4\) represents 1999. Substitute these values into the equation \(s = 476(1.13)^{t}\) to find the sales in 1997 and 1999.

Step 2: Sales in 1997

Sales in 1997 correspond to \(t=2\). Substitute this into the equation: \(s=476(1.13)^2\). Solve this to get the sales for 1997.

Step 3: Sales in 1999

Similarly, for finding sales in 1999 substitute \(t=4\), \(s=476(1.13)^4\). Solve this to get the sales for 1999.

Step 4: Find the ratio

To find the ratio of the sales in 1997 to the sales in 1999, divide the sales in 1997 by the sales in 1999. Simplify the fraction if necessary to give the final ratio.

Key Concepts

Exponential Functions

Exponential functions are a class of mathematical functions characterized by an equation where the independent variable is an exponent. These functions are typically written in the form of f(x) = ab^x, where a represents the initial value, b is the base that is raised to the power of x, and x itself is the variable exponent representing how many times the growth (or decay) is compounded.

In our example, the function s = 476(1.13)^t illustrates this beautifully. The constant 476 signifies the sales in the base year 1995, and 1.13 is the growth factor corresponding to the yearly percent increase. By plugging different values of t, which represents time passed since 1995, we can calculate the sales in any given year. This type of function is especially useful to represent growth that is not linear but accelerates over time.

Percent Increase

Percent increase is a way of expressing how much a certain value has increased as a percentage of its original value. It's commonly used in financial analyses, sales data, and other fields to represent growth over time. To calculate the percent increase, you subtract the original value from the new value, divide this by the original value, and then multiply by 100 to convert it to a percentage.

In the context of our exercise example, the number 1.13 in the exponential function represents a 13% increase each year as we can understand it from the equation's context. The way we see this is by realizing that the base b in our function, when it exceeds 1, denotes a percentage increase. Therefore, 1.13 implies a 13% yearly increase in sales.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario in order to analyze it and make predictions. These models can take many forms, including equations, functions, simulations, and more, based on the nature of what's being modeled and the relationships between different elements within the system.

The sales model s = 476(1.13)^t is an example of a mathematical model designed to predict future sales based on a known percent increase each year. It abstracts the complex dynamics of sales related to market conditions, consumer behavior, and other factors into a simple yet powerful equation. This model gives an easy-to-understand estimation of future values and can be very accurate if the percent increase remains consistent over time. However, it's important to note that these models are based on assumptions that may not always hold true, as real-world conditions can change unpredictably.