Problem 12 - Intermediate Algebra

Question

A wood sculptor carves three types of statues with a chainsaw. The number of hours required for carving, sanding, and painting a totem pole, a bear, and a deer are shown in the table. How many of each should be produced to use all available labor hours? $$ \begin{array}{|l|c|c|c|c|} \hline & \text { Totem pole } & \text { Bear } & \text { Deer } & \text { Time available } \\ \hline \text { Carving } & 2 \mathrm{hr} & 2 \mathrm{hr} & 1 \mathrm{hr} & 14 \mathrm{hr} \\ \text { Sanding } & 1 \mathrm{hr} & 2 \mathrm{hr} & 2 \mathrm{hr} & 15 \mathrm{hr} \\ \text { Painting } & 3 \mathrm{hr} & 2 \mathrm{hr} & 2 \mathrm{hr} & 21 \mathrm{hr} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Produce 3 totem poles, 2 bears, and 4 deer to use all labor hours.

Step 1: Define Variables

Let's define variables for the number of each statue produced. Let \( x \) be the number of totem poles, \( y \) be the number of bears, and \( z \) be the number of deer. We need to find the values of \( x \), \( y \), and \( z \) that will fully utilize the available labor hours without exceeding them.

Step 2: Set Up Equations from Constraints

Based on the table, we can set up the following system of linear equations:1. Carving: \( 2x + 2y + z = 14 \)2. Sanding: \( x + 2y + 2z = 15 \)3. Painting: \( 3x + 2y + 2z = 21 \)

Step 3: Simplify and Solve the System of Equations

First, let's eliminate the variable \( y \). Subtract the sanding equation from the painting equation:\[(3x + 2y + 2z) - (x + 2y + 2z) = 21 - 15\]Which simplifies to:\[2x = 6 \Rightarrow x = 3\]

Step 4: Substitute Back to Find Other Variables

Substitute \( x = 3 \) back into the carving equation:\[2(3) + 2y + z = 14 \Rightarrow 6 + 2y + z = 14 \]Solving for \( z \):\( 2y + z = 8 \).Now substitute \( x = 3 \) into the sanding equation:\[3 + 2y + 2z = 15 \Rightarrow 2y + 2z = 12 \]Solve for \( y \) and \( z \): divide by 2:\[y + z = 6 \]

Step 5: Solve the Simultaneous Equations for y and z

Now, we have two equations:1. \( 2y + z = 8 \)2. \( y + z = 6 \)Subtract the second equation from the first:\[(2y + z) - (y + z) = 8 - 6 \]\[y = 2\]Now substitute \( y = 2 \) back into \( y + z = 6 \):\[2 + z = 6 \Rightarrow z = 4\]

Step 6: Verify and Summarize Result

Now we know that \( x = 3 \), \( y = 2 \), and \( z = 4 \). Let's verify using the constraints:Carving hours: \( 2(3) + 2(2) + 4 = 6 + 4 + 4 = 14 \) ✔️Sanding hours: \( 3 + 4 + 8 = 15 \) ✔️Painting hours: \( 9 + 4 + 8 = 21 \) ✔️All constraints check out, meaning our solution is correct!

Key Concepts

Linear Equations

Linear equations are mathematical expressions that display a linear relationship between variables. In simpler terms, this means that the variables are raised to the first power and their graph forms a straight line. Linear equations are usually written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.

• Given their straightforward nature, each solution to a linear equation lies on a line in a coordinate plane.
• They are often utilized to model real-world situations where a straight-line relationship exists.

In our sculptor's exercise, each step to create the statues—carving, sanding, and painting—can be modeled through a linear equation. This is because each task takes a definitive amount of time for each type of statue. By setting up these equations, we can represent the total labor hours required as a linear system. Understanding how to work with linear equations is fundamental when dealing with systems of equations, allowing us to solve for unknown variables within certain constraints.

Constraint Solving

Constraint solving involves finding values for variables that satisfy a set of conditions or restrictions. In real life, constraints are found everywhere—be it in budget limits, time restrictions, or resource allocation, as seen in this exercise.

• When dealing with a system of linear equations, the constraints are represented as equations that the solutions must satisfy.
• The core idea is to find solution combinations that fulfill all conditions simultaneously.

In the example given, our constraint was the number of labor hours available for each task. The equations generated from the time table must match the constraints provided by the available work hours. Effectively solving constraint problems such as these requires careful setup of equations and a strategy, like substitution, to isolate and solve for each variable. Recognizing and satisfying every constraint ensures the task can be completed without any excess or shortage of resources.

Substitution Method

The substitution method is a key strategy for solving systems of equations, especially when you have two or more equations. It involves isolating one variable in one equation and substituting that expression into another equation.

• This method is particularly useful when one of the equations is easily manipulated, allowing you to express one variable in terms of another.
• It simplifies solving systems by reducing the number of variables and equations step by step.

This was applied effectively in the exercise. We started by isolating and solving for one variable—\( x = 3 \)—then proceeded to substitute this value back into the other equations to solve for \( y \) and \( z \). By systematically doing this, you reduce your complex system of equations to simpler problems that are easier to solve.The substitution method also helps verify solutions by allowing reassessment of each equation with the found values, ensuring all end values meet the initial constraints.