Problem 79 - Algebra A Combined Function

Question

The per capita consumption (in pounds) of all beef in the United States is given by the function \(C(x)=-0.33 x+67.1,\) where \(x\) is the number of years since \(2000 .\) (Source: Based on data from the Economic Research Service, U.S. Department of Agriculture, \(2000-2008\) ) a. Find and interpret \(C(4)\). b. Estimate the per capita consumption of beef in the United States in \(2010 .\)

Step-by-Step Solution

Answer

C(4) = 65.78 pounds in 2004. In 2010, consumption was estimated at 63.8 pounds.

Step 1: Understanding the Function

The function given is \(C(x) = -0.33x + 67.1\), where \(x\) represents the number of years since the year 2000. This means if \(x = 0\), it's the year 2000, if \(x = 4\), it's the year 2004, and so on. The function calculates the per capita consumption of beef in pounds for a given year based on \(x\).

Step 2: Calculating C(4)

To find \(C(4)\), substitute \(x = 4\) in the function: \[ C(4) = -0.33(4) + 67.1 \]Calculate the multiplication first: \(-0.33 \times 4 = -1.32\). Then, add this to 67.1: \[ C(4) = -1.32 + 67.1 = 65.78 \]Thus, \(C(4) = 65.78\).

Step 3: Interpretation of C(4)

The value of \(C(4) = 65.78\) represents the estimated per capita consumption of beef in pounds in the year 2004. This indicates an average of 65.78 pounds per person for that year.

Step 4: Finding the Year Corresponding to 2010

To find the number of years since 2000 for the year 2010, subtract 2000 from 2010: \[ x = 2010 - 2000 = 10 \].This indicates that \(x = 10\) for the year 2010.

Step 5: Estimating C(10) for 2010

Substitute \(x = 10\) into the function to estimate the consumption for 2010: \[ C(10) = -0.33(10) + 67.1 \]First, calculate \(-0.33 \times 10 = -3.3\). Then, add 67.1 to get the final result: \[ C(10) = -3.3 + 67.1 = 63.8 \].

Key Concepts

Per Capita Consumption

Per capita consumption refers to the average amount of a particular product consumed per person within a given population. In the context of this exercise, we are looking at beef consumption in the United States. The formula provided is a linear function, suggesting that the consumption changes at a consistent rate over time.

• The function is defined as \(C(x) = -0.33x + 67.1\), where \(x\) represents the years since 2000.
• By inputting different values of \(x\), we can determine the average beef consumption for different years.

For example, \(C(4) = 65.78\) pounds in 2004. This means that on average, each person consumed approximately 65.78 pounds of beef in 2004. Understanding this helps to gauge changes in dietary trends or economic conditions that could affect consumption patterns over time.

Function Interpretation

Function interpretation involves understanding how to read and make sense of a mathematical function. The function \(C(x) = -0.33x + 67.1\) is an example of a linear function. Here's what the components mean:

• The slope \(-0.33\) indicates the rate of change in per capita consumption. A negative slope suggests that consumption decreases by 0.33 pounds each year.
• The y-intercept \(67.1\) represents the expected consumption when \(x = 0\), or in the year 2000. It indicates the starting point of the data.

By recognizing these parts of the function, we can make predictions and conclusions about data trends over the specified time period. For instance, from year to year, if other conditions remain constant, we expect the consumption to decrease by 0.33 pounds.

Estimation in Functions

Estimation in functions allows us to predict values at certain points that are not explicitly given. By substituting desired values into our given function, we can calculate estimations for those values.For example, to estimate the consumption in 2010, we first identified \(x = 10\) as the number of years since 2000. Plugging \(x = 10\) into the function:\[ C(10) = -0.33(10) + 67.1 = 63.8 \]This calculation shows that in 2010, the estimated per capita beef consumption was 63.8 pounds per person. Using estimation in functions:

• We can project future values based on historical data, as demonstrated by extrapolating consumption trends.
• Estimation helps in planning and decision-making for resource allocation and policy development.