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Timber stumpage prices: The following table shows timber stumpage prices for pine pulpwood in two regions of the American South. \({ }^{20}\) Prices are in dollars per ton and were recorded at the start of the indicated year. $$ \begin{array}{|c|c|c|} \hline \text { Year } & \text { Southeast } & \text { Mid-Atlantic } \\ \hline 2002 & 6.50 & 4.00 \\ \hline 2005 & 7.50 & 6.70 \\ \hline 2007 & 7.00 & 10.00 \\ \hline \end{array} $$ a. For the Mid-Atlantic, what is the average rate of change per year in price from 2002 to 2005 ? b. Use your answer to part a to estimate the price in the Mid-Atlantic region at the start of \(2004 .\) (The actual price was \(\$ 6.40\) per ton.) c. For the Southeast, what is the average rate of change per year in price from 2005 to 2007 ? d. Use your answer to part c to estimate the price in the Southeast at the start of 2008 . e. For each region find the percentage increase in price from 2002 to 2007 . f. On the basis of your answer to part e, in the absence of other factors, would an investor in timber be better advised to choose the Southeast or the Mid-Atlantic?

Step by step solution

01

Calculating Average Rate of Change for Mid-Atlantic (2002-2005)

The average rate of change per year can be found using the formula: \( \text{Rate} = \frac{\text{Final Price} - \text{Initial Price}}{\text{Number of Years}} \). For the Mid-Atlantic from 2002 to 2005, the prices are \\(6.70 in 2005 and \\)4.00 in 2002. So, \( \text{Rate} = \frac{6.70 - 4.00}{2005 - 2002} = \frac{2.70}{3} = 0.90 \). The average rate of change is \$0.90 per year.

02

Estimating Price for Mid-Atlantic in 2004

To estimate the price in 2004, use the 2002 price and add the average rate multiplied by the number of years from 2002. Thus, \( \text{Price in 2004} = 4.00 + 0.90 \times 2 = 5.80 \). The estimated price is \$5.80 per ton.

03

Calculating Average Rate of Change for Southeast (2005-2007)

Using the same formula as in Step 1, calculate: For Southeast from 2005 to 2007, the prices are \\(7.00 in 2007 and \\)7.50 in 2005. Thus, \( \text{Rate} = \frac{7.00 - 7.50}{2007 - 2005} = \frac{-0.50}{2} = -0.25 \). The average rate of change is \(-\$0.25\) per year.

04

Estimating Price for Southeast in 2008

Following the method from Step 2, find the estimated price in 2008 given the 2007 price and the rate: \( \text{Price in 2008} = 7.00 - 0.25 \times 1 = 6.75 \). The estimated price is \$6.75 per ton.

05

Calculating Percentage Increase (2002-2007)

For percentage increase: \( \text{Percentage Increase} = \frac{\text{Final Price} - \text{Initial Price}}{\text{Initial Price}} \times 100 \%\). Southeast: \( \left(\frac{7.00 - 6.50}{6.50}\right) \times 100 = 7.69\%\). Mid-Atlantic: \( \left(\frac{10.00 - 4.00}{4.00}\right) \times 100 = 150\% \).

06

Advising the Investment Choice

Based on the percentage increase from 2002 to 2007, the Mid-Atlantic region experienced a 150% increase in price compared to Southeast's 7.69%. Therefore, the Mid-Atlantic is more attractive for investment based on historical data.

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Key Concepts

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

Percentage Increase

Percentage increase helps us understand how much something has grown over time. It shows the growth as a percent, making it easier to compare against other numbers. For example, let's look at the timber stumpage prices for pine pulpwood listed in the exercise. In the Southeast region, the stumpage price increased from \( \\(6.50 \) in 2002 to \( \\)7.00 \) in 2007. We can calculate the percentage increase as:

• Subtract the initial price from the final price: \( \\(7.00 - \\)6.50 = \\(0.50 \)
• Divide the change by the initial price: \( \frac{\\)0.50}{\\(6.50} \approx 0.0769 \)
• Convert to a percent by multiplying by 100: \( 0.0769 \times 100 = 7.69\% \)

This tells us that the price of pine pulpwood in the Southeast increased by approximately 7.69% over those years.
This method becomes even more illuminating when we look at the Mid-Atlantic region, which saw a substantial rise, from \( \\)4.00 \) to \( \$10.00 \), marking a 150% increase.

Price Estimation

Estimating future prices is important when making predictions or decisions based on past data. At times, it involves utilizing average rates of change to project future values.In the example provided, the average price change for the Mid-Atlantic region from 2002 to 2005 was calculated to be \( \\(0.90 \) per year. Using this information, we estimated the price for 2004 by starting from the 2002 price and adding the average change for two years. This gives:

• \( \text{Estimated Price in 2004} = \\)4.00 + 0.90 \times 2 = \$5.80 \)

Such projections are a typical part of economic modeling, helping businesses and investors anticipate future situations to make well-informed decisions. However, actual future prices can vary due to unexpected factors not accounted for in the estimation.

Investment Strategy

An effective investment strategy assesses historical data to determine future potential. Investors look at trends, such as increases or decreases in price, to guide their choices. Based on the exercise, the example calculation of the percentage increase in the Mid-Atlantic region (150%) compared to the Southeast (7.69%) would suggest a higher growth potential in the Mid-Atlantic for timber investments, assuming all other conditions remain unchanged. Investors would be enticed by the significantly higher rate of increase. When determining an investment strategy, it’s crucial to consider other factors as well, such as:

• Market stability
• Demand fluctuations
• Environmental considerations

These factors can impact price trends and should be analyzed alongside simple percentage changes when forming a comprehensive investment strategy.

Mathematical Modeling

Mathematical modeling is a process of using mathematics to represent, analyze, and predict real-world situations. In the context of economic data, like the timber stumpage prices, mathematical modeling allows for the interpretation of trends over time. The average rate of change acts as a simple yet powerful model to describe how a quantity grows or shrinks over a given period. In the problem, we see its application in determining rates of change and making future predictions, like forecasting prices for future years. These models are often created using:

• Linear equations, for straightforward relationships
• Exponential equations, when growth accelerates

By analyzing past data, mathematical models help in making informed predictions and crafting strategies for future scenarios, simplifying complex data into intelligible insights.