Problem 40 - 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. 55 miles per hour for 3 hours

Step-by-Step Solution

Answer

The distance traveled is 165 miles.

Step 1: Identify Variables

Identify the given values from the problem. Here the rate of travel, \(r\), is 55 miles per hour, and the time traveled, \(t\), is 3 hours.

Step 2: Use the Formula

Recall the formula for distance: \(D = rt\). We will use this formula to find the distance.

Step 3: Substitute the Values

Substitute the given values into the formula. Replace \(r\) with 55 miles per hour and \(t\) with 3 hours: \(D = 55 \times 3\).

Step 4: Perform the Calculation

Multiply the rate by the time to calculate the distance: \(D = 55 \times 3 = 165\).

Step 5: State the Result

The distance traveled is 165 miles.

Key Concepts

Rate and Time

When it comes to understanding how fast or how long something takes, we rely on the concepts of rate and time. Rate is essentially a way to describe speed or velocity—how quickly or slowly something happens. In our context, it refers to the speed at which someone or something travels. We usually measure it in miles per hour (mph) or kilometers per hour (km/h), depending on the country.

Time, on the other hand, measures the duration over which something occurs. The unit of time that is frequently used in problems like this is hours, but it can also be minutes or seconds. By combining the rate of speed and the time traveled, we can better understand how far something has moved.

In solving problems that involve rate and time, always ensure that the units match. If the speed is in miles per hour, the time should also be in hours for the units to work correctly.

Calculation of Distance

To calculate distance, we use the very handy and straightforward formula: \[ D = r imes t \] where \( D \) is the distance traveled, \( r \) is the rate of travel (or speed), and \( t \) is the time taken.

Here's a simple breakdown of how to use the formula:

• Identify the speed or rate of travel, denoted as \( r \).
• Determine the time \( t \) the traveler moves at that speed.
• Multiply the speed (\( r \)) by the time (\( t \)) to get the distance \( D \).

In our example, the speed is 55 mph and the time is 3 hours. Substituting these values into the formula gives us:\[ D = 55 \space ext{mph} imes 3 \space ext{hours} = 165 \space ext{miles} \] This calculation shows the total distance traveled.

Distance Traveled

Distance traveled tells us how far something or someone has moved over a period of time. It's the physical ground covered during travel, and it’s usually measured in units like miles or kilometers.

• If you're traveling at a constant speed, calculating distance is straightforward using the formula we discussed.
• If the speed varies, the calculation becomes more complex and might require additional steps.

In our exercise example, a car travels 55 miles each hour over 3 hours. Therefore, the distance traveled is simply 55 times 3, resulting in 165 miles. This gives us a clear and concise understanding of how far the vehicle has moved from its starting point, at constant speed.