Problem 39 - Elementary Algebra

Question

The distance traveled \(D\) is equal to the average rate \(r\) times the time traveled \(t\) at that rate: \(D=r t .\) Determine the distance traveled given the rate and the time. 60 miles per hour for 3 hours

Step-by-Step Solution

Answer

The distance traveled is 180 miles.

Step 1: Identify Given Values

Identify the average rate (\(r\)) and the time traveled (\(t\)) from the problem. Here, \(r = 60\) miles per hour and \(t = 3\) hours.

Step 2: Apply the Distance Formula

Use the formula for distance traveled, \(D = rt\), where \(D\) is the distance traveled, \(r\) is the average rate, and \(t\) is the time.

Step 3: Calculate the Distance

Substitute the given values into the formula: \(D = 60 \times 3\). Perform the multiplication to find that \(D = 180\).

Key Concepts

Understanding Rate

The rate in this context refers to how fast something is moving or the speed of an object. It's a measure of distance covered per unit of time. When we say the rate is 60 miles per hour, it means for every hour traveled, 60 miles are covered.
This concept is crucial because it tells us how much ground is covered over a period of time. In the distance formula, the rate is represented by the letter \( r \). Understanding rate allows us to predict how far we will go over a certain timeframe given a consistent speed.- Rate is represented in units like miles per hour, kilometers per hour, etc.- It stays constant in simple problems like the one we're solving.- It gives a clear picture of the object's speed to help calculate distance.

Defining Time Traveled

Time traveled refers to the duration an object has been moving at a particular rate. In our problem, it's the time spent traveling at the given speed. Understanding how long something travels helps us to use the distance formula effectively.Time is often represented by the letter \( t \) in the formula \( D = rt \). Knowing the time is essential because it multiplies with the rate to determine the total distance.- Time is usually measured in hours, minutes, or seconds.- Real-world scenarios might require conversions between these units.- It helps determine changes in motion or the total distance covered.

The Role of Multiplication in the Distance Formula

Multiplication is the mathematical operation used in the distance formula \( D = rt \). This operation allows us to find the total distance by combining both the rate and the time.In the given problem, we multiply the rate (60 miles per hour) by the time (3 hours) to find the distance traveled. - Multiplication combines two numbers to calculate the product.- It ensures both factors (rate and time) contribute to the final result.- Always perform the operation carefully to get the correct answer.