Problem 38 - Algebra A Combined Function

Question

The Pentagon is the world's largest office building in terms of floor space. It has three times the amount of floor space as the Empire State Building. If the total floor space for these two buildings is approximately 8700 thousand square feet, find the floor space of each building.

Step-by-Step Solution

Answer

Empire State Building: 2175 thousand sq ft, Pentagon: 6525 thousand sq ft.

Step 1: Understanding the Problem

We are given that the Pentagon's floor space is three times that of the Empire State Building. Together, they total 8700 thousand square feet. We need to find the floor space of each building.

Step 2: Setting Up an Equation

Let the floor space of the Empire State Building be \( x \). Then, the floor space for the Pentagon will be \( 3x \) because it is three times the Empire State Building's floor space. The total floor space is the sum of these two: \( x + 3x = 8700 \) thousand square feet.

Step 3: Simplifying the Equation

Combine the terms on the left side of the equation: \( x + 3x = 4x \). Therefore, the equation simplifies to \( 4x = 8700 \).

Step 4: Solving for the Empire State Building

Divide both sides of the equation by 4 to solve for \( x \): \( x = \frac{8700}{4} \). This results in \( x = 2175 \) thousand square feet.

Step 5: Calculating the Pentagon's Floor Space

Since the Pentagon's floor space is three times that of the Empire State Building, multiply \( x \) by 3: \( 3 \times 2175 = 6525 \). Therefore, the Pentagon's floor space is 6525 thousand square feet.

Key Concepts

Variables

Understanding variables is essential in solving algebra word problems. Variables are symbols or letters, often represented by letters like \( x \) or \( y \), used to stand in for unknown values in mathematical expressions and equations. In our exercise, we use a variable to represent the floor space of the Empire State Building. We call this variable \( x \). This method allows us to set up a relationship with what we know about the Pentagon's floor space, which is three times that of the Empire State Building. Using variables helps us keep track of these relationships and simplifies the problem by turning words into algebraic expressions that are easier to manipulate.

Equations

Equations are mathematical statements showing that two expressions are equal. In problem-solving, they provide a way to represent relationships and conditions described in word problems. Here, an equation is formulated using the given information that the total floor space of the Pentagon and the Empire State Building is 8700 thousand square feet.

• We know: the floor space of the Pentagon is three times that of the Empire State Building.
• So, the equation will be: \( x + 3x = 8700 \).

This equation allows us to express the total floor space in terms of the variable \( x \), which stands for the Empire State Building's space. By setting up an equation, we have a structured way to find the unknown variable and, consequently, deduce more information.

Problem Solving

Problem solving in algebra involves organizing and breaking down the information given in a word problem to find a solution. First, we must understand the problem, then represent it with a variable, and set up an equation. To solve:

• Identify the unknowns and establish variables, as we did with \( x \) for the Empire State Building's floor space.
• Translate the relationships into equations.
• Use algebraic techniques to simplify the equations and solve for the variable.

Once we have the value of the variable, we can use it to infer additional information requested by the problem, like the floor space of the Pentagon. Each step in the process builds on the previous one, using logic and arithmetic to find solutions.

Mathematical Operations

Mathematical operations represent the procedures we can perform on numbers and variables to solve equations. Basic operations include addition, subtraction, multiplication, and division. In this exercise, we see several operations in play.- **Addition**: Used to combine the floor spaces of the two buildings \( x + 3x \).- **Simplification**: We simplify the left side of the equation: \( 4x = 8700 \).- **Division**: We divide both sides of the equation by 4 to find the value of \( x \): \( x = \frac{8700}{4} = 2175 \).- **Multiplication**: To find the Pentagon's floor space, we multiply the Empire State Building's floor space by 3: \( 3 \times 2175 = 6525 \).These operations allow us to manipulate equations and solve for the unknowns, ultimately arriving at the solution.