Problem 70 - Elementary and Intermediate Algebra

Question

Translate each problem to an equation. Do not solve. Travel to Work. For U.S. cities with populations greater than \(5000,\) the longest average commute is 59.8 min in Indian Wells, Arizona. This is \(51.2 \mathrm{min}\) longer than the shortest average commute, which is in Fort Bliss, Texas. How long is the average commute in Fort Bliss?

Step-by-Step Solution

Answer

\( 59.8 = x + 51.2 \)

Step 1: Identify the Given Information

The problem states that the longest average commute is 59.8 minutes in Indian Wells, Arizona. It also mentions that this is 51.2 minutes longer than the shortest average commute in Fort Bliss, Texas.

Step 2: Define the Variable

Let’s define the variable: \( x \) - the average commute time in Fort Bliss, Texas.

Step 3: Set Up the Equation

According to the problem, the longest commute (59.8 minutes) is 51.2 minutes longer than the shortest commute in Fort Bliss. This relationship can be written as: \[ 59.8 = x + 51.2 \]

Key Concepts

average commute time

The concept of average commute time is an important measure in many studies related to urban planning and quality of life. It represents the typical amount of time that people spend traveling to work. In our exercise, two cities are compared based on their average commute times: Indian Wells, Arizona, and Fort Bliss, Texas.
Indian Wells has the longest average commute time among U.S. cities with populations greater than 5000, clocking in at 59.8 minutes. Understanding average commute times helps in assessing the efficiency of transportation systems and can highlight areas needing infrastructure improvements.

defining variables

Defining variables is a crucial step in translating word problems into equations. Variables act as placeholders for unknown values we need to find.
In this example, we need to find the average commute time in Fort Bliss, Texas. We choose a variable, let's say \( x \), to represent this unknown commute time. By doing so, we make it easier to set up and solve an equation that models the relationship between the given pieces of information and the unknown quantity. Defining \( x \) at the start helps streamline the problem-solving process and ensures clarity in our calculations.

setting up equations

Setting up equations from word problems is the next logical step after defining variables. The given information must be translated into a mathematical statement that represents the problem accurately.
For instance, the problem states that the longest average commute time (59.8 minutes in Indian Wells) is 51.2 minutes longer than the shortest average commute (in Fort Bliss). With our variable \( x \) defined as the shortest average commute time, we can set up the following equation: \[ 59.8 = x + 51.2 \] Here, we see how the relationships described in the word problem are represented mathematically. The 'longest commute' is broken down into the 'shortest commute' plus an extra 51.2 minutes. This equation can later be solved to find the value of \( x.\) Remember, the goal in this step is to accurately model the relationship using correct mathematical expressions.