Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Organic Food Sales Organic food sales in the United States in millions of dollars \(x\) years past 2005 can be modeled by \(O(x)=2649.4 x+13,260\) (a) Evaluate \(O(9)\) and interpret your result. (b) Use the formula for \(O(x)\) to write an equation that gives the organic food sales \(y\) during year \(x\). (c) Refer to part (b) and find \(y\) when \(x=2014\). (d) Use your equation in part (b) to determine the year when organic food sales reached \(\$ 26,507\) million.

Step-by-Step Solution

Answer

(a) $37,104.6 million in 2014. (b) $y = 2649.4(x - 2005) + 13,260$. (c) $37,104.6 million. (d) Year 2010.

Step 1: Evaluate O(9)

To evaluate \(O(9)\), substitute \(x = 9\) into the given formula \(O(x) = 2649.4x + 13,260\). This calculates the organic food sales 9 years past 2005. \[ O(9) = 2649.4 \times 9 + 13,260 = 23,844.6 + 13,260 = 37,104.6 \]The result, approximately \(37,104.6\) million dollars, represents the organic food sales in 2014.

Step 2: Write the Equation for Organic Sales

To write an equation for organic food sales \(y\) as a function of the year \(x\), note that the base year is 2005. Therefore, if \(x\) is the actual year, the number of years past 2005 is \(x - 2005\). Substitute \(x - 2005\) into the equation for \(O(x)\):\[ y = 2649.4(x - 2005) + 13,260 \] This is the equation for organic food sales \(y\) during year \(x\).

Step 3: Find y when x=2014

Substitute \(x = 2014\) into the equation from Step 2. \[ y = 2649.4(2014 - 2005) + 13,260 \] \[ y = 2649.4 \times 9 + 13,260 = 23,844.6 + 13,260 = 37,104.6 \]The organic food sales in 2014 were approximately \(37,104.6\) million dollars.

Step 4: Determine the Year for Sales of $26,507 million

We need to find the year when \(y = 26,507\). Substitute \(y = 26,507\) into the equation from Step 2 and solve for \(x\):\[ 26,507 = 2649.4(x - 2005) + 13,260 \] Subtract \(13,260\) from both sides:\[ 13,247 = 2649.4(x - 2005) \] Divide both sides by \(2649.4\):\[ x - 2005 = \frac{13,247}{2649.4} \approx 5 \]Add 2005 to both sides to find \(x\):\[ x = 2005 + 5 = 2010 \] Organic food sales reached \(26,507\) million dollars in the year 2010.

Key Concepts

Modeling with Equations

Modeling with equations is a fundamental concept in mathematics used to describe relationships between variables. In this exercise, we are modeling organic food sales in the United States. The model is given by the equation:

• \( O(x) = 2649.4x + 13,260 \)

This equation is a linear function, where \(x\) represents the number of years past 2005. The term \(2649.4x\) is the slope, representing the rate of change in sales. It shows how much the sales increase per year. The constant \(13,260\) is the y-intercept, showing the sales level at the base year (2005).
This model allows us to predict sales for any year by plugging in the number of years past 2005 into \(x\). We can solve for different values to find out how sales have grown over the years, providing useful insights into market trends.

Evaluating Functions

Evaluating functions involves calculating the value of a function for a specific input. This step is crucial in understanding how a model works and what it represents. To evaluate \(O(9)\), we substitute \(9\) into the equation:

• \( O(9) = 2649.4 \times 9 + 13,260 \)

When we do the calculations: - \(2649.4 \times 9\) equals approximately \(23,844.6\).
- Adding \(13,260\) results in \(37,104.6\) million dollars.
This tells us the predicted sales for the year 2014. Evaluating functions allows you to apply a mathematical model to real-world scenarios by obtaining tangible results and predictions.

Interpreting Results

Interpreting the results is essential after evaluating a function. It helps put the numbers in context, providing meaning and insight. From the calculation \(O(9) = 37,104.6\), we interpret that in 2014, the organic food sales were approximately \(37,104.6\) million dollars.
This is compared to earlier or later years to analyze growth trends. For instance, if the sales were significantly lower in 2005 but increased to that level by 2014, it would indicate a substantial industry growth. Thus, interpreting these results can aid stakeholders in making informed decisions about the organic food market.

Solving for Variables

Solving for variables in an equation is about finding unknown values that make the equation true. In part (d) of the exercise, we determine the specific year when sales reach \(26,507\) million dollars.
Here's how we solve for \(x\):

• Use the equation: \( 26,507 = 2649.4(x - 2005) + 13,260 \)
• Subtract \(13,260\) from both sides to isolate the terms with \(x\):
  \(13,247 = 2649.4(x - 2005)\)
• Divide both sides by \(2649.4\) to find \((x - 2005) = \frac{13,247}{2649.4} \approx 5\)
• Add 2005 to solve for \(x\): \(x = 2005 + 5 = 2010\)

This process shows how algebraic manipulation can determine when a certain condition, such as a sales target, is met.