Question

The revenue \(R\) (in millions of dollars per year) for Papa John's from 1996 to 2005 can be modeled by \(R=\frac{-485.0+116.68 t}{1-0.12 t+0.0097 t^{2}}, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: Papa John's Int'l.) (a) During which year, from 1996 through 2005 , was Papa John's revenue the greatest? the least? (b) During which year was the revenue increasing at the greatest rate? decreasing at the greatest rate? (c) Use a graphing utility to graph the revenue function, and confirm your results in parts (a) and (b).

Short answer

The answer will vary based on calculations performed using the function. To get the exact years of maximum and minimum revenue, as well as years of highest increase and decrease rates, perform steps 1 to 4 as described.

Step-by-step solution

Calculate Revenue for Each Year

First, plug in the values of \( t \) from 6 to 15 (representing the years 1996 to 2005) into the given equation \( R=\frac{-485.0+116.68 t}{1-0.12 t+0.0097 t^{2}} \) to find the revenue for each year.

Determine Maximum and Minimum Revenue

Next, identify the highest and lowest revenue values from the results obtained in Step 1. The corresponding year to these values will indicate the years with maximum and minimum revenue.

Calculate the Derivative of R

To determine the rate at which revenue is increasing or decreasing, calculate the derivative of \( R \) with respect to \( t \). The derivative, \( R'(t) \), represents the rate of change of revenue concerning time.

Determine Maximum and Minimum Rate of Change

Evaluate \( R'(t) \) for \( t \) from 6 to 15. The maximum value will indicate the year with the greatest rate of increase in revenue, and the minimum value will indicate the year with the greatest rate of decrease in revenue.

Graph the Function

Graphically represent the function \( R(t) \) using a graphing utility. The peaks and valleys of the function will be visually representative of the maximum and minimum points of revenue and the rate of change we've calculated previously. Verify your results.

Key concepts

Derivative Calculations

In understanding how revenues change for a company like Papa John's over a specific period, calculating derivatives plays a crucial role. The derivative represents the rate at which the revenue, a function of time here, is changing at any given moment. For a given revenue function, such as \( R = \frac{-485.0 + 116.68 t}{1 - 0.12 t + 0.0097 t^2} \), the derivative with respect to \( t \) is denoted as \( R'(t) \).

To find this derivative:

• Apply the quotient rule, given that \( R \) is a ratio of two functions.
• The quotient rule states that the derivative of \( \frac{u}{v} \) is \( \frac{u'v - uv'}{v^2} \), where \( u \) and \( v \) are differentiable functions of \( t \).

By calculating \( R'(t) \), we find how fast the revenue increases or decreases in different years. This information is crucial for assessing the company’s performance and making informed decisions in business planning.

Rate of Change

The rate of change in the context of this exercise tells us how quickly the revenue for Papa John's is increasing or decreasing between the years 1996 and 2005. By examining the derivative \( R'(t) \), we can identify the periods of rapid growth, stability, or decline.

Here's how you approach it:

• After determining \( R'(t) \), plug in values of \( t \) from 6 to 15.
• Observe the sign of \( R'(t) \). A positive value indicates increasing revenue, while a negative value indicates declining revenue.

The year where \( R'(t) \) reaches its maximum is when the revenue was growing at the fastest pace. Conversely, the year when \( R'(t) \) hits its lowest value denotes the fastest decline in revenue.

Understanding these rates helps businesses plan adequately for the future, allocating resources effectively during growth periods or making austerity measures during declines.

Graphing Functions

Graphing the revenue function allows us to visually comprehend the fluctuations in revenue over the years. By plotting \( R(t) = \frac{-485.0 + 116.68 t}{1 - 0.12 t + 0.0097 t^2} \), we can pinpoint key years with maximum or minimum revenues and observe the overall trend.

The process involves:

• Using a graphing utility to input the function and set the domain for \( t \) from 6 to 15.
• In analyzing the graph, look out for peaks and troughs. Peaks represent maximum revenue years, while troughs show minimum revenue years.
• Additionally, observe the slope of the graph over time. Steeper slopes correspond to rapid changes in revenue, whether increase or decrease.

Graphing not only confirms the results derived from calculations but also offers an intuitive understanding of revenue trends, which is invaluable for strategic business decisions.