Problem 52 - A Graphical Approach to Precalculus with Limits

Question

A glue company needs to make some glue that it can sell for \(\$ 120\) per barrel. It wants to use 150 barrels of glue worth \(\$ 100\) per barrel, along with some glue worth \(\$ 150\) per barrel and glue worth \(\$ 190\) per barrel. It must use the same number of barrels of \(\$ 150\) and S190 glue. How much of the \(\$ 150\) and \(\$ 190\) glue will be needed? How many barrels of \(\$ 120\) glue will be produced

Step-by-Step Solution

Answer

30 barrels of each \( \$150 \) and \( \$190 \) glue are needed, producing 210 barrels of \( \$120 \) glue.

Step 1: Define Variables

Let \( x \) be the number of barrels of \( \\(150 \) glue and \( y \) be the number of barrels of \( \\)190 \) glue. The number of barrels of \( \\(150 \) and \( \\)190 \) glue must be equal, so \( x = y \). We also want to produce barrels of glue to be sold for \( \$120 \) each.

Step 2: Set Up the Equation for Total Glue

Since we're using 150 barrels of \( \\(100 \) glue, \( x \) barrels of \( \\)150 \) glue, and \( y \) barrels of \( \$190 \) glue, the total number of barrels produced will be \( 150 + x + y \).

Step 3: Set Up the Revenue Equation

We need the average price to be \( \$120 \). The total cost becomes \( 100(150) + 150x + 190y \). The equation for the price per barrel is:\[ \frac{100(150) + 150x + 190y}{150 + x + y} = 120 \]

Step 4: Solve the Equation

Replacing \( y \) with \( x \) (since \( x = y \)), we have the equation:\[ \frac{100(150) + 150x + 190x}{150 + x + x} = 120 \]Simplifying gives:\[ \frac{15000 + 340x}{150 + 2x} = 120 \]Cross-multiplying, we have:\[ 15000 + 340x = 120(150 + 2x) \]\[ 15000 + 340x = 18000 + 240x \]Subtracting \( 240x \) from both sides:\[ 100x = 3000 \]Dividing by 100, we find:\[ x = 30 \]

Step 5: Calculate Number of Barrels Produced

Since \( x = y \), we know the number of barrels more: \( x + y = 60 \). Therefore, the total number of barrels produced is \( 150 + x + y = 150 + 60 = 210 \).

Key Concepts

Variables Definition

To solve the linear equation word problem related to production costs in a glue company, the first step is to define the variables clearly. This is crucial for setting up the equations that will help us find the solution. In this particular problem, we are dealing with three types of glue with different costs per barrel. We define:
\( x \) as the number of barrels of glue worth \( \\( 150 \) per barrel
\( y \) as the number of barrels of glue worth \( \\) 190 \) per barrel
It's important to note that the problem specifies the constraint that the number of barrels for both \( \\( 150 \) and \( \\) 190 \) glue must be equal, hence, \( x = y \). These variables form the basis of our mathematical model.

Revenue Equation

Setting up a correct revenue equation is critical for solving any financial linear equation problem. In this case, we're looking to create glue that has a total production cost averaging \( \\(120 \) per barrel. To do that, we have to account for all types of glue used:

• 150 barrels of \( \\)100 \) glue
• \( x \) barrels of \( \\(150 \) glue
• \( y \) barrels of \( \\)190 \) glue

The revenue equation captures the balance between these components. It is expressed as:
\[ \frac{100(150) + 150x + 190y}{150 + x + y} = 120 \]
This equation tells us that the total cost of all barrels, when divided by the total number of barrels, must equal the target price of \( \$120 \) per barrel. Understanding and being able to derive such an equation is essential, as it perfectly encapsulates the financial balance aimed for by the glue company.

Cross-multiplication

Cross-multiplication is a useful mathematical technique, often employed in solving rational equations and proportions. Especially in problems requiring clearing denominators, such as our revenue equation. The expression:
\[ \frac{15000 + 340x}{150 + 2x} = 120 \]
is simplified using cross-multiplication, leading to a new form that is easier to resolve. This means multiplying each side by the denominator of the other side to eliminate the fraction:
\[ 15000 + 340x = 120(150 + 2x) \]
By applying cross-multiplication, we align terms in a straightforward linear form ready for basic algebraic manipulation. This makes resolving the variable much simpler.

Problem Solving Steps

Conquering linear equation word problems often requires a systematic problem-solving approach. Breaking down the solution into distinct steps enhances clarity and aids understanding. Here are the steps followed in this problem:

• Define Variables: Identify what unknowns you need to solve by establishing variables, noting any constraints or equalities, such as \( x = y \).
• Set Up Equations: Use the problem statement to establish equations. This involves both unit equations ('barrels produced') and goal equations (the 'revenue equation').
• Simplify Equations: Reduce complex equations using mathematical rules like substitution and cross-multiplication.
• Solve and Conclude: Resolve the equations to find the value of the variables, and perform necessary calculations.

Following these steps will methodically guide students to a solution, while also ensuring they understand each part of the problem.