Problem 40 - Thinking Mathematically

Question

Traveling at an average rate of between 40 and 50 miles per hour for 3 to 4 hours, select the best estimate for the distance traveled. a. 120 miles b. 160 miles c. 195 miles d. 210 miles

Step-by-Step Solution

Answer

The best estimate for the distance traveled is 195 miles as it lies within the calculated range (between 120 and 200 miles).

Step 1: Calculate the minimum distance

Firstly, calculate the minimum distance scenario by using the lesser values in the provided ranges. That is, if the speed is 40 miles per hour and the time is 3 hours. This is done by multiplying these two values together, resulting in \(40 * 3 = 120\) miles.

Step 2: Calculate the maximum distance

Next, calculate the maximum distance scenario by using the larger values in the provided ranges. Here, this would mean considering the speed as 50 miles per hour and the time as 4 hours. By multiplying these values together, the result will be \(50 * 4 = 200\) miles.

Step 3: Select the closest estimate

Having found both the minimum and maximum possible distances, the next step is to select the estimate that fits within this range. In this case, the options are: 120 miles (a), 160 miles (b), 195 miles (c), or 210 miles (d). The distance should be between 120 and 200 miles, so the best estimate would be option (c) 195 miles.

Key Concepts

Rate of Speed

The rate of speed is essentially how fast an object is moving, and it is a fundamental concept in estimating distance traveled. It is usually expressed in terms of units of distance covered per unit of time, like miles per hour (mph) or kilometers per hour (kph).

In everyday situations, the rate of speed helps you determine how long it will take to reach a destination if you know the distance, or vice versa. It’s a logical way to set expectations for travel time and to plan journeys. When looking at the textbook problem, the provided speeds range from 40 to 50 mph. This information is critical, as different rates of speed will result in different travel distances for the same amount of time.

Time Duration

Time duration measures the amount of time something takes or lasts. For travel, it's the portion of time spent moving from one location to another. Generally, it is measured in seconds, minutes, hours, or a combination of these units.

The significance of time duration in our textbook exercise is that it complements the rate of speed to formulate an estimate for distance traveled. Given a specific speed, if the duration increases, the distance also increases proportionately. In the example problem, we're given a duration ranging from 3 to 4 hours. Variations in this time frame will affect the distance estimate as clearly shown in the step-by-step solution.

Multiplication of Rates and Time

The multiplication of rate and time is an operation that calculates distance. It combines the concepts of rate of speed and time duration to yield the total distance traveled. The basic formula for this calculation is Distance = Rate x Time, often abbreviated as D = RT.

This equation allows us to perform estimations with varying speeds and times. As in the exercise, by multiplying the lowest speed with the shortest time, we get the minimum distance. Conversely, the highest speed multiplied by the longest time frame gives us the maximum possible distance. The actual travel distance should fall between these two calculated extremes. This technique is fundamental in solving problems related to travel and motion, and it’s the cornerstone of our textbook solution.