Problem 19 - 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. \(^{8}\) Write an equation relating \(a\) and \(r\) if the plantation yields \(1,000,000 \mathrm{~kg}\) of coffee.

Step-by-Step Solution

Answer

Question: Find the equation that relates the hectares of Arabica (a) and hectares of Robusta (r) based on their yields and the given total production. Answer: The equation that relates the hectares of Arabica and Robusta is 5a + 8r = 6666.67.

Step 1: Determine the yield of each variety

We first need to determine how much coffee is produced from each variety. - For Arabica, we multiply the hectares (a) by the yield per hectare of Arabica (750 kg): 750a - For Robusta, we multiply the hectares (r) by the yield per hectare of Robusta (1200 kg): 1200r

Step 2: Combine the yields to match the total production

Now that we have the amounts produced by both Arabica and Robusta, we will add them together, since this sum should equal the total production of 1,000,000 kg. Therefore, we get the equation: 750a + 1200r = 1,000,000

Step 3: Simplify the equation (if possible)

The given equation is 750a + 1200r = 1,000,000, and we can see that all the coefficients are divisible by 150. Dividing all terms of the equation by 150 will make the equation easier to work with: (750a)/150 + (1200r)/150 = (1,000,000)/150 5a + 8r = 6666.67 So, the simplified equation relating the hectares of Arabica and Robusta is: 5a + 8r = 6666.67

Key Concepts

Understanding Linear Equations

Linear equations are equations of the first order. An equation like this represents a straight line when plotted on a graph. The general form of a linear equation is \[ ax + by = c \] where:

• \(a\), \(b\), and \(c\) are constants.
• \(x\) and \(y\) are variables.

In our coffee plantation exercise, the linear equation \[ 5a + 8r = 6666.67 \] is used. Here, \(a\) and \(r\) are the variables representing hectares of Arabica and Robusta respectively, while \(5\) and \(8\) are coefficients that relate to the yield per hectare.
Linear equations like this are fundamental in algebra because they are the basis for analyzing and solving many real-world problems.
By learning how to manipulate and solve these equations, you develop skills that allow you to work with more complex algebraic structures. Remember, the key is in balancing the equation so that both sides remain equal.

Applying Problem Solving Techniques

Problem solving in mathematics often involves certain strategies, such as identifying what information is given and what needs to be found. In our exercise:

• We have been given the yield per hectare for Arabica and Robusta.
• We need to find a relationship between the hectares planted of each to achieve a total yield of 1,000,000 kg.

This process involves translating the problem into a mathematical form, which in this case is a linear equation.
First, identify the yield from each type of coffee plant:

• Arabica: \(750a\)
• Robusta: \(1200r\)

Next, combine them to form an equation that sums up to the total yield: \[ 750a + 1200r = 1,000,000 \]
Simplifying this makes it easier to interpret and solve, as shown in our solution with the equation \[ 5a + 8r = 6666.67 \].
Throughout this process, clarity in each step ensures the problem is efficiently solved.

Exploring Agricultural Mathematics

Agricultural mathematics involves using mathematical models to solve problems related to agriculture. These problems can involve optimizing resources like land or predicting the outcomes based on different farming strategies.
In the context of our exercise, we are optimizing land allocation for two types of coffee plants to meet a specific production goal.

• Determining how many hectares should be dedicated to Arabica and Robusta to achieve a yield of 1,000,000 kg.
• Using equations to model potential yields because resources are finite and must be efficiently used.

This exercise reflects real agricultural decisions where mathematical calculations help to maximize production while minimizing costs and ensuring sustainable farming practices.
Understanding concepts like linear equations aids in making such decisions, providing a clearer picture of how changes in one area (e.g., the number of hectares allocated) can affect overall outcomes.