Problem 53 - Introductory Algebra for College Students

Question

If you are traveling in your car at an average rate of \(r\) miles per hour for \(t\) hours, then the distance, \(d,\) in miles, that you travel is described by the formula \(d=r t\) : distance equals rate times time. a. Solve the formula for \(t\) b. Use the formula in part (a) to find the time that you travel if you cover a distance of 100 miles at an average rate of 40 miles per hour.

Step-by-Step Solution

Answer

The time taken to travel a distance of 100 miles at an average speed of 40 miles per hour is 2.5 hours.

Step 1: Rearrange the formula to solve for \(t\)

The original formula is \(d = r \cdot t\). Since the task is to solve for \(t\), we should divide both sides of the equation by \(r\) to get \(t= \frac{d}{r}\). So, time (\(t\)) is equal to distance (\(d\)) divided by rate (\(r\)).

Step 2: Substituting into the formula

Given that the distance (\(d\)) covered is 100 miles and the average speed (\(r\)) is 40 miles per hour, replace \(d\) and \(r\) with 100 and 40 respectively in the formula \(t= \frac{d}{r}\). So, \(t = \frac{100}{40}\).

Step 3: Calculating the time

To get the time, perform the division operation mentioned in the previous step. Therefore, \(t = 2.5\) hours.

Key Concepts

Solving Equations

When you are dealing with mathematical problems, solving equations is a common task. It's like being a detective, finding the value for a variable, like 't' in our exercise. The equation given in the exercise is part of the distance, rate, and time formula family, expressed as \(d = r \cdot t\). However, our goal is to find the time (\(t\)).
To solve for \(t\), rearrange the formula. You want \(t\) all by itself on one side of the equation. Here’s how you can do it:

• Start with the initial equation \(d = r \cdot t\).
• Since \(t\) is multiplied by \(r\), you'll do the opposite operation (division) to isolate \(t\).
• Divide both sides by \(r\) to isolate \(t\), getting \(t = \frac{d}{r}\).

Voila! You've successfully solved the equation for time. By practicing these rearrangements, you can easily adapt when other values in similar equations need finding.

Travel Problems

Travel problems often involve figuring out distances, speedd, or time based on some given information. These problems can feel like mini-puzzles, where you need to use the right pieces to find a solution. In this exercise, we focused on determining the time of travel, a common scenario in travel problems.
The steps involved were:

• Using given information like the distance covered (100 miles) and the average speed (40 miles per hour).
• Plugging these values into the formula for time, \(t = \frac{d}{r}\).
• Solving for \(t\) gives you the total time taken for the journey.

This method helps in practical situations, like planning road trips or estimating arrival times. Travel problems demonstrate real-world applications of math, making abstract concepts more tangible.

Average Speed

Average speed is a fundamental concept in travel problems, representing how much distance you cover in a certain amount of time. It's calculated by dividing the total distance by the total time taken. If you’re constantly changing your speed during travel, the average speed gives you a balanced view of your journey's pace.
In our exercise:

• We know the average speed (\(r\)) was 40 miles per hour.
• This speed allows us to calculate the time when given a set travel distance.
• The formula \(t = \frac{d}{r}\) utilizes the average speed to determine the time taken for a given distance (100 miles in our problem).

Understanding average speed is useful beyond math class. It determines how we plan daily commutes or long trips, ensuring efficiency and timely arrivals. By mastering how to calculate and apply average speed, you make travel planning less daunting and more precise.