Question

Revenue The revenue \(R\) (in millions of dollars per year) for Symantec Corporation from 1997 through 2005 can be modeled by \(R=\frac{1340 t^{2}+24,044 t+22,704}{-6 t^{2}+94 t+100}\) where \(t=7\) corresponds to 1997 . Find the total revenue from 1997 through \(2005 .\) Then find the average revenue during this time period. (Source: Symantec Corporation)

Short answer

To calculate the total revenue and average revenue, the appropriate values of \(t\) ranging from 7 to 15 (representing the years 1997 through 2005, respectively) should be substituted into the given function, and the respective values of \(R\) are computed. These values represent the yearly revenues. The total revenue is then calculated by summing up these yearly revenues. Lastly, the average revenue is found by dividing the total revenue by the number of years, which is 9.

Step-by-step solution

Step 1

Understand the given function. The revenue \(R\) for Symantec Corporation is given by the function \(R=\frac{1340t^{2}+24,044t+22,704}{-6t^{2}+94t+100}\), where \(t=7\) corresponds to 1997. This implies that whenever the year is given, the corresponding value of \(t\) can be found by subtracting 1990 from the given year.

Step 2

Find the yearly revenues from 1997 through 2005. This can be done by substituting the respective values of \(t\) (which are 7 to 15) into the given equation of \(R\) and calculating the respective revenues.

Step 3

Calculate the total revenue from 1997 through 2005. This is done by summing up all the yearly revenues found in Step 2.

Step 4

Determine the average revenue during this time period. The average revenue is found by dividing the total revenue (from step 3) by the number of years, which in this case, is 9.

Key concepts

Rational Functions

Rational functions are a type of function formed by the ratio of two polynomials. In revenue modeling, they are useful for representing complex relationships. For Symantec Corporation, the revenue model provided is a rational function. The numerator and the denominator are distinct polynomial functions of the variable \( t \). Rational functions can show growth or declines in revenue as \( t \) changes. The Rational function \( R=\frac{1340t^{2}+24,044t+22,704}{-6t^{2}+94t+100} \) is designed to model revenue over time.

• Numerator: Dictates how the revenue scales with \( t \). In our model, it includes the quadratic term \( 1340t^2 \), indicating that revenue changes more significantly due to \( t \) values.
• Denominator: Affects how the polynomial suppresses or boosts revenue as \( t \) changes. It tempers the effects of changes to the numerator.

Understanding rational functions helps in predicting how changes in variables like time might affect overall revenue.

Average Revenue

Average revenue provides an insightful measure for financial analysis, giving an idea of the "middle" performance level over a time span. Calculating this is crucial when evaluating a company's stability and growth potential. On a fundamental level, average revenue is the total income divided by the number of periods examined. In the specific case of Symantec, once the total revenue for each year between 1997 and 2005 is known, you calculate the average as follows:

• Sum up all the yearly revenues.
• Count the number of years involved.
• Divide the total revenue by the number of years to find the average.

Computing the average revenue helps businesses to:

• Understand typical financial performance over time.
• Make informed decisions about future planning and investment.

Average revenue gives a consistent point of reference that simplifies interpretation of the data.

Symantec Corporation Revenue

Understanding the revenue trends of organizations like Symantec Corporation is vital for grasping industry dynamics and company performance. The given equation models the revenue in million dollars from 1997 through 2005. Here, each change in \( t \) equates to a year, starting with \( t = 7 \) for 1997.Companies often use such equations to represent revenue over multiple years. This is part of a broader financial analysis, offering insights into bouts of high revenue or potential downswings. By plugging in values of \( t \) from the year span, you can derive each year's revenue. This data enables stakeholders to:

• Track historical performance.
• Identify patterns or anomalies in financial behavior.
• Strategize based on past performance trends.

These historical insights are key for Symantec’s strategic business decisions and future revenue projections.

Yearly Revenue Calculation

When calculating yearly revenue, the goal is to utilize the rational function to convert each year into a corresponding value of \( t \) and find its revenue output. For Symantec, from 1997 to 2005, \( t \) ranges from 7 to 15. Plug each value into the revenue function:\[ R = \frac{1340t^{2}+24,044t+22,704}{-6t^{2}+94t+100} \]This function helps derive specific yearly revenue figures:1. Start with \( t = 7 \) for 1997. Plug it into the function.2. Repeat the process incrementally until you reach \( t = 15 \) for 2005.3. Calculate each result using basic algebra.Collect all revenues over the period to derive total earnings. Summing these results gives the total revenue for 1997 through 2005. It enables an overview of the company's financial trajectory across the span, useful for evaluating business health and growth.