Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. The Coffee Shoppe sells a coffee blend made from two coffees, one costing $$\$ 5 / \mathrm{lb}$$ and the other costing $$\$ 6 / \mathrm{lb}$$. If the blended coffee sells for $$\$ 5.60 / \mathrm{lb}$$, find how much of each coffee is used to obtain the desired blend. Assume that the weight of the blended coffee is \(100 \mathrm{lb}\).

Step-by-Step Solution

Answer

To create the desired 100 lbs coffee blend priced at $5.60 per lb, we need to mix 40 lbs of the first type of coffee (\(5 per lb) and 60 lbs of the second type of coffee (\)6 per lb).

Step 1: Assign variables to the problem

Let x represent the amount of the first type of coffee (\(5 per lb) and y represent the amount of the second type of coffee (\)6 per lb). The total weight of the blend is 100 lbs.

Step 2: Set up the system of linear equations

Based on the information given, we can form two equations: 1. The sum of the amounts of the two types of coffee should equal 100 lbs: \(x + y = 100\) 2. The weighted average price of the blend should be $5.60 per lb: \(5x + 6y = 5.60 \times 100\)

Step 3: Solve for one variable using one equation

Let's solve for y in the first equation: \(y = 100 - x\)

Step 4: Substitute the solution from step 3 into the second equation

We can now rewrite the second equation by substituting the solution for y from step 3: \(5x + 6(100-x)= 5.60 \times 100\)

Step 5: Solve for x

Simplify and solve for x: \(5x + 600 - 6x = 560\) \(-x = -40\) \(x = 40\)

Step 6: Solve for y using the value of x

We can now substitute the value of x back into the equation we found in step 3: \(y = 100 - x = 100 - 40 = 60\)

Step 7: Interpret the result

The result indicates that 40 lbs of the first type of coffee (\(5 per lb) and 60 lbs of the second type of coffee (\)6 per lb) should be mixed together to achieve the desired blend with a selling price of $5.60 per lb and a total weight of 100 lbs.

Key Concepts

System of Equations

In the problem of blending two types of coffee with different prices, we employ a system of linear equations to find a solution. A system of equations consists of multiple equations that share variables. By solving them simultaneously, we find values for the variables that satisfy all the equations simultaneously.

For our coffee blend problem, let's break it down:

• We have two variables: let \(x\) represent the pounds of coffee costing \(5 per pound and \(y\) represent the pounds of coffee costing \)6 per pound.
• Our first equation is derived from the total weight of the blend: \(x + y = 100\).
• The second equation comes from the blend's price requirement: \(5x + 6y = 560\).

By using both equations, we can determine the exact amounts of each type of coffee needed to meet both the weight and price conditions. This method of using a system of equations is a powerful way to handle problems involving multiple constraints with shared variables.

Weighted Average

A weighted average is a type of average where each component is multiplied by a factor representing its importance, or weight, before the final sum is calculated. In our coffee problem, the concept of weighted average is used to determine the price of the coffee blend. This ensures that the final cost per pound takes into account the different prices of the coffee components.

Here's how the weighted average applies:

• Each pound of \(5 coffee is considered, and its cost is factored in by multiplying by 5.
• Similarly, each pound of \)6 coffee is considered by multiplying by 6.
• The formula for the weighted average cost is then expressed as \(5x + 6y\), where \(x\) and \(y\) are the pounds of each type.
• We need the weighted average of the blend to be $5.60 for each of the 100 pounds, which gives us the equation \(5x + 6y = 560\).

By solving this relationship alongside the total weight equation, you ensure the average price reflects the contribution of each coffee type correctly, achieving the desired blend price.

Problem Formulation

Problem formulation is the essential process of defining and understanding the problem before solving it. This involves identifying the knowns, unknowns, and conditions that govern the scenario. When approaching the coffee blend problem, proper formulation allows for a clear path towards the solution.

To effectively formulate this problem:

• Identify the goal: create a blend of coffee that meets a specific weight and price constraint.
• Define variables: choose variables that will help describe the parts of the blend—here, \(x\) and \(y\) were used.
• Construct equations: use the given conditions (total weight and blend price) to set up equations that describe the situation.

Formulating the problem in this structured way ensures that nothing is missed and provides a logical foundation for solving the equations using algebraic methods. With proper formulation, solutions unfold more clearly, minimizing errors and streamlining the process.