Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Blending coffees A shop specializes in preparing blends of gourmet coffees. From Colombian, Costa Rican, and Kenyan coffees, the owner wishes to prepare 1-pound bags that will sell for \(\$ 12.50\). The cost per pound of these coffees is \(\$ 14, \$ 10\), and \(\$ 12\), respectively. The amount of Colombian is to be three times the amount of Costa Rican. Find the amount of each type of coffee in the blend.

Step-by-Step Solution

Answer

Costa Rican: 0.125 lbs, Colombian: 0.375 lbs, Kenyan: 0.5 lbs.

Step 1: Assign Variables

Let's assign variables to represent the pounds of each type of coffee in the blend. Let \( x \) be the pounds of Costa Rican coffee, \( 3x \) be the pounds of Colombian coffee (since it's three times the Costa Rican), and \( y \) be the pounds of Kenyan coffee.

Step 2: Write the Equation for Total Weight

Since the blend is prepared into 1-pound bags, the total weight of all the coffees in the blend must equal 1 pound. Therefore, the equation is: \[ x + 3x + y = 1 \] Simplifying, this gives: \[ 4x + y = 1 \]

Step 3: Write the Equation for Total Cost

The owner needs to use the cost of coffees to determine how to reach the selling price of \( \$ 12.50 \). The cost equation based on each type's price per pound is: \[ 10x + 14(3x) + 12y = 12.5 \] Simplifying gives: \[ 10x + 42x + 12y = 12.5 \] \[ 52x + 12y = 12.5 \]

Step 4: Solve the System of Equations

Now we have two equations: \[ 4x + y = 1 \] and \[ 52x + 12y = 12.5 \]. First, solve the first equation for \( y \): \[ y = 1 - 4x \]. Substitute this in the second equation: \[ 52x + 12(1 - 4x) = 12.5 \]. Solve this equation: \[ 52x + 12 - 48x = 12.5 \] \[ 4x + 12 = 12.5 \] \[ 4x = 0.5 \] \[ x = 0.125 \].

Step 5: Calculate Other Coffee Amounts

Using \( x = 0.125 \), find \( y \) and the Colombian coffee amount: \[ y = 1 - 4x = 1 - 4(0.125) = 0.5 \] and \[ 3x = 3(0.125) = 0.375 \]. The blend contains 0.125 pounds of Costa Rican coffee, 0.375 pounds of Colombian coffee, and 0.5 pounds of Kenyan coffee.

Key Concepts

Systems of Equations

A system of equations helps solve problems where there are multiple unknowns. In this coffee blending problem, we have three types of coffee: Colombian, Costa Rican, and Kenyan. We need to determine how much of each coffee is in a blend that weighs exactly 1 pound and meets a specific cost.

To tackle such problems, we use variables. Here, we set:

• Let \( x \) be the pounds of Costa Rican coffee.
• Since the Colombian coffee is three times that of the Costa Rican, it is represented as \( 3x \).
• Let \( y \) be the pounds of Kenyan coffee.

By using these variables, we form two equations with two unknowns. These equations are solved simultaneously to find the values of \( x \) and \( y \), providing us with the desired mix of coffees.

Cost Analysis

Cost analysis allows us to find the ideal mix of coffees that totals a specific cost. Each coffee has a different price, affecting the blend's overall cost. In this problem, the owner wants to sell the coffee blend at \( \\(12.50 \).

The cost per pound of each coffee is:

• Colombian: \( \\)14 \)
• Costa Rican: \( \\(10 \)
• Kenyan: \( \\)12 \)

To ensure the blend hits the desired selling price, we create a cost equation. This equation calculates the total cost by multiplying the amount of each coffee by its price per pound, then adding them up to match the selling price of \( \$12.50 \). This helps determine the feasible amounts of each coffee type to reach the target cost.

Algebraic Modeling

Algebraic modeling allows us to take a real-world scenario and express it in mathematical terms, leading to equations that can be solved. In our coffee blending problem, this involves both weight and cost aspects.

We started by establishing the total weight equation: \( 4x + y = 1 \). This represents the total weight of Costa Rican, Colombian, and Kenyan coffees combined, all equaling 1 pound.

Then, the cost equation: \( 52x + 12y = 12.5 \). This models the requirement that the total cost of the blend matches the desired selling price.
By setting up these equations, algebraic modeling helps translate the practical ideas into manageable mathematical expressions, which we then solve to find the blend ratios.

Mathematical Reasoning

Mathematical reasoning provides a logical approach to solving problems through a step-by-step process. It involves analyzing the problem, forming equations, and using algebraic techniques to find solutions.

After establishing our systems of equations, mathematical reasoning guides us to:

• Solve the first equation for one variable in terms of others. Here, we solved for \( y \) as \( y = 1 - 4x \).
• Substitute this expression into the other equation to eliminate one variable, making it easier to solve.
• Perform algebraic operations to isolate and calculate the unknown values.

This logical flow ensures a structured solution, enabling us to determine the quantities of each coffee in the blend logically and accurately.