91Ó°ÊÓ

Polo Ralph Lauren Revenue The revenue (in million dollars) of the Polo Ralph Lauren Corporation from 2002 through 2009 is given in the table. a. Use the data to estimate the year in which revenue was growing most rapidly. b. Find a model for the data. c. Give the first derivative for the model in part \(b\) with units. d. Determine the year in which revenue was growing most rapidly. Find rate of change of revenue in that year. $$ \begin{aligned} &\text { Polo Ralph Lauren Corporation Annual Revenue }\\\ &\begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Revenue } \\ \text { (billion dollars) } \end{array} \\ \hline 2002 & 2.36 \\ \hline 2003 & 2.44 \\ \hline 2004 & 2.65 \\ \hline 2005 & 3.31 \\ \hline 2006 & 3.75 \\ \hline 2007 & 4.30 \\ \hline 2008 & 4.88 \\ \hline 2009 & 5.02 \\ \hline \end{array} \end{aligned} $$

Step by step solution

01

Analyze Revenue Growth

To find the year in which the revenue was growing most rapidly, we first calculate the annual growth rate. This is done by subtracting the previous year's revenue from the current year's revenue for each year in the table.

02

Record Growth Differences

Using the data, calculate the difference in revenue for each year: - 2003: 2.44 - 2.36 = 0.08 billion - 2004: 2.65 - 2.44 = 0.21 billion - 2005: 3.31 - 2.65 = 0.66 billion - 2006: 3.75 - 3.31 = 0.44 billion - 2007: 4.30 - 3.75 = 0.55 billion - 2008: 4.88 - 4.30 = 0.58 billion - 2009: 5.02 - 4.88 = 0.14 billion. The highest increase is in 2005.

03

Find a Model for the Data

Using a regression tool or a graphing calculator, fit the data to a model. A quadratic model might better fit the revenue growth:\[ f(t) = at^2 + bt + c \] where \( t \) represents the year (with 2002 as \( t = 0 \)).

04

Derive the Revenue Growth Rate

Using the quadratic function from Step 3, find its first derivative. The derivative represents the rate of change of revenue over time:\[ f'(t) = 2at + b \]. The units of this derivative would be billion dollars per year, as it represents change in revenue per change in year.

05

Determine Maximum Revenue Growth

Set the derivative from Step 4 equal to zero to find the critical points:\[ 2at + b = 0 \]. Solve for \( t \) to determine when revenue growth was maximal. Given the derivative, solve for \( t \) using the parameters from your model.

06

Calculate Rate of Change at Maximum

Substitute the value of \( t \) found in Step 5 back into the derivative \( f'(t) \) to find the rate of change at that year. This gives the maximum rate of revenue growth in billion dollars per year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Revenue Growth Analysis

Revenue growth analysis is an essential tool in economics to assess how a company's revenue progresses over time. By analyzing yearly revenue changes, we can identify patterns and peak growth periods. This involves:

• Calculating the difference between consecutive years' revenues.
• Identifying the year with the maximum increase compared to previous years.

For instance, in the case of Polo Ralph Lauren Corporation's annual revenue from 2002 to 2009, we notice the most significant growth occurred between 2004 and 2005. This year experienced a revenue increase of 0.66 billion dollars.
Effectively understanding revenue growth trends can inform strategic business decisions, such as investment in marketing, expansion, or product development. Analyzing these patterns over multiple years helps businesses to forecast future earnings more accurately and make informed strategic choices.

Quadratic Models

Quadratic models are mathematical representations used to describe data that may follow a parabolic trend. To fit a quadratic model to data, we generally use the form:\[f(t) = at^2 + bt + c\] where:

• \( t \) represents the time variable (years, in this case, starting from a base year such as 2002).
• \( a, b, \) and \( c \) are coefficients determined through regression analysis.

Quadratic models are particularly useful in economics for representing revenue growth over time due to their ability to capture acceleration and deceleration in growth. By using regression analysis, you can determine the best-fitting quadratic function to match historical data points.
In the context of Polo Ralph Lauren, this model allows us to predict future revenues and understand how revenue growth evolves. It captures initial growth periods and potential slowdowns or speed-ups in economic performance.

First Derivative in Economic Models

The first derivative of a function, often referred to as the rate of change, is crucial for analyzing economic models. For a quadratic revenue model, the first derivative is:\[f'(t) = 2at + b\]This derivative provides insight into how quickly revenue is changing at a particular point in time. The units are typically in billion dollars per year, reflecting the change in revenue over the change in time.
In economics, finding when the derivative equals zero (\( f'(t) = 0 \)) is vital because it identifies points of maximum or minimum growth rates.

• When the derivative is positive, revenue is increasing.
• When zero, it indicates potential peak growth or a transition point.
• If negative, it implies decreasing revenue trends.

This analysis is invaluable for businesses as it allows them to pinpoint when they were most efficiently generating revenue, enabling strategic adjustments to maintain or enhance growth.