Problem 39 - Thinking Mathematically

Question

Traveling at an average rate of between 60 and 70 miles per hour for 3 to 4 hours, select the best estimate for the distance traveled. a. 90 miles b. 190 miles c. 225 miles d. 275 miles

Step-by-Step Solution

Answer

The best estimate for the distance traveled is 190 miles (option b) as it falls within the range calculated.

Step 1: Determine minimum distance

Use the minimum values provided for speed and time to calculate the minimum distance. This is done by multiplying the speed (60 miles per hour) by the time (3 hours). Thus, the minimum distance = 60 miles/hour * 3 hours = 180 miles.

Step 2: Determine maximum distance

Use the maximum values provided for speed and time to calculate the maximum distance. This is done by multiplying the speed (70 miles per hour) by the time (4 hours). Thus, the maximum distance = 70 miles/hour * 4 hours = 280 miles.

Step 3: Select the best estimate for distance

Now, having calculated the minimum (180 miles) and maximum (280 miles) possible distances, look at the provided answer choices. The best estimate would be the one that falls within this range.

Key Concepts

Understanding Rate of Speed

The rate of speed is a crucial concept when it comes to estimating travel distances. It's essentially how fast something is moving over a particular period of time, typically expressed in miles per hour (mph) or kilometers per hour (kph). In the given exercise, we talk about the rate of speed in terms of 60 to 70 mph. This range helps us understand how fast a vehicle can travel under varying conditions. For example:

• If a car is moving at a slower speed within the range, like 60 mph, it will cover less ground compared to its faster counterpart.
• On the other hand, at a higher speed such as 70 mph, a vehicle can traverse a greater distance in the same time span.

It's essential to remember that speed rate distribution isn't always constant due to factors like traffic or road conditions, meaning the average speed may fluctuate between the given range. Understanding and estimating the rate of speed is foundational for making accurate distance calculations.

Simple Time Calculation

Time calculation is another key part of solving distance problems. Time, in this context, is simply the period over which the object is moving. In our example, the car travels anywhere from 3 to 4 hours. This range provides a window of time during which the journey might take place. Calculating distance traveled requires pairing this time frame with a rate of speed. Key points to consider:

• Shorter travel time means less distance covered if speed remains constant.
• Longer travel time allows more distance to be covered at the same speed.

By experimenting with different time values within the range and multiplying them by known speeds, we can estimate potential distances. However, time alone won't provide distance without considering speed, which must always be taken into conjunction with time for meaningful calculations.

Estimating Distance Range

Distance range estimation involves finding the span between the minimum and maximum distances possible within given constraints. In this exercise, the distances were calculated using both the highest and lowest parameters for speed and time. Here's how it all works:

• The minimum possible distance is calculated by multiplying the lowest speed (60 mph) by the shortest time frame (3 hours), giving 180 miles.
• Conversely, the maximum distance uses the highest speed (70 mph) and longest time frame (4 hours), resulting in 280 miles.

Understanding this range is crucial when we need to choose the best estimate among several options. Answer choices that fall outside this range can be swiftly eliminated. This method ensures that the estimate reflects realistic travel scenarios, which are affected by speed and time variations during the journey. Estimating a distance range not only helps in practical travel planning but also aids in exercises like this to reach sensible and accurate conclusions.