Problem 20 - Algebra Form and Function

Question

The coffee variety Arabica yields about \(750 \mathrm{~kg}\) of coffee beans per hectare, while Robusta yields about 1200 \(\mathrm{kg} /\) hectare. In Problems \(19-20\), suppose that a plantation has a hectares of Arabica and \(r\) hectares of Robusta. \(^{8}\) On August \(14,2003,\) the world market price of coffee was \(\$ 1.42 / \mathrm{kg}\) for Arabica and \(\$ 0.73 / \mathrm{kg}\) for Robusta. \(^{9}\) Write an equation relating \(a\) and \(r\) if the plantation produces coffee worth \(\$ 1,000,000\).

Step-by-Step Solution

Answer

Question: Given a plantation that produces Arabica and Robusta coffee, the yield per hectare for Arabica is 750 kg and the price per kilogram is $1.42. The yield per hectare for Robusta is 1200 kg with a price of $0.73 per kilogram. If the plantation's total earnings from both varieties amount to $1,000,000, find the equation that relates the hectares of Arabica coffee (a) to the hectares of Robusta coffee (r). Answer: a * (750 * 1.42) + r * (1200 * 0.73) = 1,000,000

Step 1: Determine the earnings from one hectare of Arabica

The earning from one hectare of Arabica coffee is the yield per hectare multiplied by the price per kilogram. In this case, Arabica yields 750 kg per hectare and the price is $1.42 per kg. Therefore, the earnings for one hectare of Arabica is 750 * 1.42.

Step 2: Determine the earnings from one hectare of Robusta

Similarly, the earning from one hectare of Robusta coffee is the yield per hectare multiplied by the price per kilogram. Robusta yields 1200 kg per hectare and the price is $0.73 per kg. Therefore, the earnings for one hectare of Robusta is 1200 * 0.73.

Step 3: Calculate the total earnings from Arabica and Robusta

Now, we will calculate the total earnings based on the number of hectares of Arabica (a) and Robusta (r). For Arabica, the total earnings would be a * (750 * 1.42) and for Robusta, the total earnings would be r * (1200 * 0.73).

Step 4: Set up the equation relating a and r

Since the plantation produces coffee worth $1,000,000, the total earnings from Arabica and Robusta will equal this amount. Hence, we can set the equation as follows: a * (750 * 1.42) + r * (1200 * 0.73) = 1,000,000

Key Concepts

Coffee Yield

Understanding coffee yield is essential in the context of agriculture and economics. Coffee yield refers to the amount of coffee beans produced per unit area, such as a hectare. Different coffee varieties offer varying yields, influenced by factors like climate, soil quality, and farming techniques. For example, the Arabica variety, known for its aromatic flavor, typically yields around 750 kg of coffee beans per hectare. In contrast, Robusta, often used for its higher caffeine content and disease resistance, yields approximately 1200 kg per hectare.
The yield not only impacts the total production but also affects the profitability and planning of coffee plantations.
Understanding the yield helps farmers and agricultural businesses make informed decisions about which variety to plant to optimize output and profitability.

Hectares

Hectares are a metric unit of area measurement commonly used in agriculture to denote large land areas. One hectare is equal to 10,000 square meters or about 2.47 acres.
In agricultural economics, knowing the number of hectares used for different crops is crucial for planning and analysis.

• Helps in calculating the yield: By understanding the area under cultivation, the total expected yield of a crop can be determined, which is vital for forecasting production levels.
• Budgeting and resources allocation: Knowing the hectares allows for the effective allocation of resources, such as labor and capital, and helps in estimating costs related to plantation maintenance.

It is important for farmers to keep track of the hectares to ensure optimal use of their land and to align their production with market demands.

Market Price

Market price is a critical concept in agricultural economics, representing the current price at which a commodity is being bought and sold. For coffee, the market price can fluctuate due to various factors such as demand, supply, global economic conditions, and even weather patterns.
In our exercise, the market price on August 14, 2003, was $1.42 per kg for Arabica and $0.73 per kg for Robusta coffee.

• Determines profitability: The market price directly affects the profitability of a coffee plantation as it determines the revenue generated from the sale of coffee beans.
• Affects planting decisions: Farmers may choose to plant more of a variety that offers a better price to ensure higher returns.
• Influences international trade: Countries that export coffee must remain competitive by understanding and adapting to market price changes.

Therefore, keeping track of market prices is essential for farmers and agricultural businesses to make sound economic decisions.

Agricultural Economics

Agricultural economics blends the understanding of economic principles with agricultural production and land management. It examines how to best allocate resources like land, labor, and capital for agricultural production to achieve maximum efficiency and profitability.
Several key aspects include:

• Production economics: Focuses on optimizing the production process and improving the efficiency of resource use within agriculture, thereby maximizing yield and profit.
• Market dynamics and pricing: Studies how market forces affect farm product prices and how farmers can adapt to these changes to remain profitable.
• Policy analysis: Evaluates the impact of governmental policies on the agriculture sector, including subsidies, tariffs, and trade agreements.

By applying economic theory to agriculture, farmers and policymakers can better understand market trends, make data-driven decisions, and improve the sector's overall sustainability and growth. Agricultural economics plays a vital role in ensuring food security and the economic viability of rural communities.