Problem 20 - 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. Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 192 units is planned, and the number of family units (two- and three-bedroom townhouses) will equal the number of one-bedroom units. If the number of one-bedroom units will be 3 times the number of threebedroom units, find how many units of each type will be in the complex.

Step-by-Step Solution

Answer

In the apartment complex, there will be \(96\) one-bedroom units, \(64\) two-bedroom units, and \(32\) three-bedroom units.

Step 1: Total number of units

There are 192 units in total: x + y + z = 192.

Step 2: Family units equal one-bedroom units

The number of family units (townhouses) is equal to the number of one-bedroom units: y + z = x.

Step 3: One-bedroom units is 3 times three-bedroom units

The number of one-bedroom units is three times the number of three-bedroom units: x = 3z. Now we'll solve these equations step by step.

Step 4: Substitute for x in the second equation

We have x = 3z, so we can replace x in the second equation: y + z = 3z.

Step 5: Solve for y

Subtract z from both sides of the equation: y = 2z.

Step 6: Substitute for x and y in the first equation

Now we can substitute x = 3z and y = 2z in the first equation: (3z) + (2z) + z = 192.

Step 7: Solve for z

Combine the terms containing z: 6z = 192. Divide both sides by 6: z = 32.

Step 8: Calculate values for x and y

Using the z value we found above, we can calculate the values for x and y using x = 3z and y = 2z, respectively. x = 3z = 3(32) = 96 y = 2z = 2(32) = 64 So there will be 96 one-bedroom units, 64 two-bedroom units, and 32 three-bedroom units in the apartment complex.

Key Concepts

Systems of Equations

When tackling problems like the Cantwell Associates' apartment complex, we utilize systems of equations. These are sets of equations with multiple variables that interrelate.
They allow us to model complex real-world scenarios mathematically and solve them by finding values for each variable.
In the context of Cantwell's problem, our variables are the number of each type of unit: `x` for one-bedroom units, `y` for two-bedroom townhouses, and `z` for three-bedroom townhouses.Let’s understand the equations formed:

• The first equation, \(x + y + z = 192\), captures the total number of units planned—an essential starting point in many modeling problems.
• The second equation, \(y + z = x\), shows the equality between family units and one-bedroom units.
• The third equation, \(x = 3z\), expresses the relationship where the number of one-bedroom units is three times that of the three-bedroom units.

By setting up these equations, we frame the logical relationships and constraints of the problem in a mathematical form. Understanding systems of equations helps us translate real-world conditions into mathematical language that we can solve.

Algebraic Expressions

Algebraic expressions are vital in forming the building blocks of our system of equations.
They represent relationships using variables, constants, and arithmetic operations.
In the Cantwell Associates' example, each term in our equations, like `3z`, represents an algebraic expression.Algebraic expressions serve multiple purposes:

• They clearly express mathematical relationships, such as the relation between one-bedroom and three-bedroom units in \(x = 3z\).
• Allow manipulation and simplification. For instance, solving \(y + z = 3z\) leads us to discover \(y = 2z\), simplifying further problem-solving.
• Facilitate substitution in systems of equations, enabling step-by-step problem solving.

By dealing with these algebraic expressions, we break down complex relationships into understandable parts.
This makes solving the problem manageable, leading to a clear and precise solution.

Problem Formulation

Problem Formulation is the initial and crucial step in solving any mathematical modeling problem.
It involves identifying the problem, defining variables, and setting up relevant equations.
For Cantwell Associates, formulation began with understanding what the problem asks: finding unit numbers for each type in a planned development. Follow these steps to effectively formulate problems:

• Identify what needs to be solved—in our case, numbers of each unit type.
• Define variables representing unknown quantities, such as `x`, `y`, and `z` in the problem.
• Translate given information and conditions into equations, connecting these variables. We used conditions like the total number of units and specific relationships between them (e.g., family units equal one-bedroom units).

By properly formulating the problem, we set a clear path for solving.
Effective formulation ensures we have all necessary information in a structured, solvable form.
This step lays the foundation for finding reliable and accurate solutions.