Problem 43 - 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 kilometers per hour for 10 hours

Step-by-Step Solution

Answer

The distance traveled is 600 kilometers.

Step 1: Identify Given Values

We are given the average rate \( r \) as 60 kilometers per hour and the time \( t \) as 10 hours. We need to use these values in our calculations.

Step 2: Apply the Formula for Distance

The formula for distance is \( D = r \, t \). To find the distance, we substitute the given values into this formula.

Step 3: Substitute the Values into the Formula

Substitute \( r = 60 \) kilometers per hour and \( t = 10 \) hours into the formula: \( D = 60 \times 10 \).

Step 4: Calculate the Distance

Perform the multiplication: \( 60 \times 10 = 600 \).

Step 5: Conclusion

The distance traveled is 600 kilometers.

Key Concepts

Distance Calculation

Calculating distance is a fundamental concept that helps us understand how far an object travels over time at a certain speed. We have a simple step-by-step method using the formula \( D = rt \), where \( D \) is the distance, \( r \) is the rate or speed, and \( t \) is time. It's important to understand that

• Distance measures how much ground is covered, usually given in units like kilometers or miles.
• Rate refers to how fast an object is moving, typically expressed in kilometers per hour (km/h) or miles per hour (mph).
• Time is the duration for which the object is moving, generally measured in hours, minutes, or seconds.

To calculate distance, simply multiply the rate by the time. This process converts the speed you are moving at for a certain time into how far you have actually traveled.
In the example provided, using a rate of 60 kilometers per hour and a time of 10 hours, we calculate the distance by substituting these values into the formula: \( D = 60 \times 10 \). Performing the multiplication gives us the distance of 600 kilometers.

Rate and Time

Understanding the relationship between rate and time is crucial when calculating distance.Rate, often described as speed, indicates how fast something is moving. It is the distance covered per unit of time and is typically found in units like km/h or mph. When driving a car at 60 km/h, it implies that in one hour, you cover 60 kilometers if you maintain a constant speed.
Time measures the period over which the movement occurs. The longer you keep moving, the more distance you will cover. For example, traveling at a constant speed for 10 hours means that the total distance is determined by how long you continue traveling.Combining rate and time provides us the tool needed to compute distance, applying them into the simple formula \( D = rt \). By plugging in the values you know, either the speed or the time, you can effortlessly find the missing piece – the distance traveled.

Step-by-Step Solution

A structured approach, or step-by-step solution, is a powerful method to break down complex problems into manageable parts. It ensures clarity and precision in mathematical computations, like distance calculation.1. **Identify Given Values**: Start by clearly noting the values you have. For our example, these are the rate \( 60 \) km/h and time \( 10 \) hours.
2. **Apply the Distance Formula**: Use the equation \( D = rt \) which helps tie together the rate of speed and the time traveled to find the distance.
3. **Substitute the Values**: Insert the known values into the formula. This step helps you focus on the calculation by using the precise numbers from the problem – here \( D = 60 \times 10 \).
4. **Calculate the Distance**: Perform the arithmetic operation. Multiplying 60 by 10 tells us the distance is 600 kilometers.
5. **Draw a Conclusion**: Clearly express the result. The final state was that the object traveled 600 kilometers. This methodical approach strengthens your problem-solving skills by ensuring every element of the problem is considered and applied correctly.