Question

From 1994 to 1999, the sales for a chain of home furnishing stores increased by about the same annual rate. The sales \(S\) (in millions of dollars) in year \(t\) can be modeled by \(S=455\left(\frac{13}{10}\right)^{t}\) where \(t\) represents years since 1994 . Find the ratio of 1999 sales to 1995 sales.

Short answer

The ratio of 1999 sales to 1995 sales is \(\left(\frac{13}{10}\right)^{4}\).

Step-by-step solution

Identify the 't' for each year

The value of 't' represents years since 1994. So, for the year 1995, \(t = 1995 - 1994 = 1\). And for the year 1999, \(t = 1999 - 1994 = 5\).

Calculate sales for each year

Now we can substitute each value of 't' into \(S = 455\left(\frac{13}{10}\right)^{t}\) to find the sales. The sales for 1995, \(S_{1995} = 455\left(\frac{13}{10}\right)^{1}\). And the sales for 1999, \(S_{1999} = 455\left(\frac{13}{10}\right)^{5}\).

Find the ratio of sales

The ratio of 1999 sales to 1995 sales is \(\frac{S_{1999}}{S_{1995}} = \frac{455\left(\frac{13}{10}\right)^{5}}{455\left(\frac{13}{10}\right)^{1}}\). After simplifying, this ratio is \(\left(\frac{13}{10}\right)^{4}\).

Key concepts

Exponential Functions

Exponential functions are a type of algebraic function where the variable appears in the exponent. For example, the formula for calculating compound interest, population growth, and as in our exercise, sales growth, often take the form of exponential functions. They are characterized by the formula:

\[ S = a(b)^{t} \]

where \( S \) represents the final amount, \( a \) is the initial amount, \( b \) is the base or growth factor, and \( t \) denotes time or the number of periods. When \( b > 1 \), the function represents exponential growth, and when \( b < 1 \), it represents exponential decay. What's essential to understand here is that in every consecutive time period, the increase is proportional to the current value, which is why it's also known as compound growth. In the case of the home furnishing stores' sales, the growth is over years, and the base, \( \frac{13}{10} \), indicates a 30% annual increase. Therefore, the exponential function models how quickly sales are increasing each year.

Modeling Real-World Situations Algebraically

Mathematical models are vital tools in understanding, describing, and predicting behaviors in real-world situations. Algebraic models, like the exponential function we've been discussing, translate real-world problems into mathematical language. To build a model, we identify key variables, establish their relationships, and write an algebraic expression or equation representing that relationship.

For example, the assigned task represents a situation where the relationship between time and sales is modeled. The formula \( S= 455\left(\frac{13}{10}\right)^{t} \) was derived based on historical data that shows sales increment trends. These models are powerful because they allow us to predict future values or compare different time frames, as long as the pattern of growth stays consistent. Additionally, by using algebraic models, we can understand the underlying mechanics of phenomena, which, in a business context, can guide strategic planning and decision-making.

Ratios

Ratios are a way to express the relationship between two numbers or quantities, showing how much of one thing there is compared to another. In the context of exponential functions, ratios are particularly useful as they can simplify complex relationships and allow for easier comparison between values at different times.

The exercise provided requires finding the ratio between the sales in 1999 and those in 1995. To do this, we used the respective formula for each year derived from the exponential function, substituting the time variable \( t \) with the number of years since 1994. After substituting and simplifying, we're left with the ratio as an exponent power of the base, \( \left(\frac{13}{10}\right)^4 \). This not only indicates the multiple of growth but also provides a clear, single numerical value that describes the relative increase in sales over the four-year interval. Ratios like this are particularly helpful when looking to quickly compare the scale of change or progress between different points in time.