Question

From 1994 to 1998 , the sales for a national clothing store increased by about the same percent each year. The sales \(S\) (in millions of dollars) for year \(t\) can be modeled by $$ S=3723\left(\frac{6}{5}\right)^{t} $$ where \(t=0\) corresponds to \(1994 .\) Find the ratio of 1998 sales to 1995 sales.

Short answer

The ratio of sales in 1998 to 1995 is \(\left(\frac{6}{5}\right)^3\).

Step-by-step solution

Find Sales Value for 1995

Substitute \(t=1\) into the model equation to get the sales for the year 1995. The result will be \(S = 3723 \left(\frac{6}{5}\right)^1 = 3723 \times \frac{6}{5}\)

Find Sales Value for 1998

Substitute \(t=4\) into the model equation to get the sales for the year 1998. The result will be \(S = 3723 \left(\frac{6}{5}\right)^4 = 3723 \times \left(\frac{6}{5}\right)^4\)

Compute the ratio

The ratio of 1998 sales to 1995 sales would be calculated by dividing sales of 1998 by that of 1995, which would be \(\frac{3723 \times \left(\frac{6}{5}\right)^4}{3723 \times \frac{6}{5}}\)

Simplify

Simplify the equation: The ratio of 1998 sales to 1995 sales will get simplified because the \(3723\) in the numerator and the denominator cancel out and \(\frac{6}{5}\) in the denominator cancels out one of the \(\frac{6}{5}\) in the numerator. Thus, the ratio simplifies to \(\left(\frac{6}{5}\right)^3\)

Key concepts

Exponential Functions

Exponential functions describe scenarios where a quantity grows or decays at a rate proportional to its current value. In the given exercise, the sales for a national clothing store are modeled by an exponential function. The formula presented, \( S = 3723\left(\frac{6}{5}\right)^t \), is a classic example of an exponential function.
This function entails a base \( \frac{6}{5} \) raised to a power \( t \), which represents time in years. The constant 3723 signifies the sales amount at the starting point, corresponding to \( t = 0 \) or the year 1994. As \( t \) increases, the exponentiation by \( \frac{6}{5} \) causes the sales values to increase exponentially, a natural way to represent cases where each year's growth builds upon the last.
Understanding exponential functions is essential for interpreting how fast-growing phenomena, like populations, investments, or in this case, sales, evolve over time.

Mathematical Modeling

Mathematical modeling involves representing real-world situations with mathematical formulas, making it possible to analyze and predict behavior quantitatively. The sales model \( S = 3723\left(\frac{6}{5}\right)^t \) serves as a mathematical model for the store's increasing sales. Such a model is powerful because it provides a simplified version of reality with considerable predictive capability.
In this model, the exponential function captures the essence of the sales growth, assuming it follows a consistent pattern over the years. Mathematical modeling helps in decision-making and strategic planning because it allows businesses to forecast future scenarios and adjust their strategies accordingly. A model is usually established based on past data, and, as is the case here, it becomes a tool for making predictions based on that past behavior extrapolated into the future.

Ratio and Proportion

Ratio and proportion are mathematical concepts that describe the relationship between quantities and their comparative measurements. In the context of this exercise, we see ratio being applied to compare sales figures from different years. Specifically, we are asked to find the ratio of sales in 1998 to those in 1995.
A ratio is a way to show how one number relates to another, and here we are interested in how much larger (or smaller) the sales numbers in 1998 are when compared to those in 1995. By computing \( \frac{S_{1998}}{S_{1995}} \), which simplifies to \( \left(\frac{6}{5}\right)^3 \), we get the proportion, which tells us that the sales in 1998 were \( \left(\frac{6}{5}\right)^3 \) times the sales in 1995. Understanding ratios and proportions is crucial in various fields, including finance, engineering, and economics because it allows comparisons that are crucial for analysis and decision-making processes.