Problem 39 - Calculus

Question

Give an explanation for your answer. If a car is going 50 miles per hour at 2 pm and 60 miles per hour at 3 pm, then it travels between 50 and 60 miles during the hour between 2 pm and 3 pm.

Step-by-Step Solution

Answer

The car travels between 50 and 60 miles between 2 pm and 3 pm, which is correct.

Step 1: Understand the Problem

To solve this problem, we aim to find out how far the car travels between 2 pm and 3 pm while traveling at varying speeds of 50 mph at 2 pm and 60 mph at 3 pm. We need to determine if the statement that the car travels between 50 and 60 miles during that hour is true.

Step 2: Determine Minimum Distance

To find the minimum distance traveled, consider the possibility that the car travels at the lower speed of 50 mph for the entire hour. Therefore, the minimum distance covered is given by:\[ \text{Minimum Distance} = 50 \text{ miles/hour} \times 1 \text{ hour} = 50 \text{ miles} \]

Step 3: Determine Maximum Distance

For the maximum distance, assume the car travels at the higher speed of 60 mph for the entire hour. Therefore, the maximum distance covered is calculated as:\[ \text{Maximum Distance} = 60 \text{ miles/hour} \times 1 \text{ hour} = 60 \text{ miles} \]

Step 4: Conclusion

Based on the calculations, the car travels a distance between the minimum of 50 miles and the maximum of 60 miles during the hour between 2 pm and 3 pm. Therefore, the statement that the car travels between 50 and 60 miles in that hour is correct.

Key Concepts

Average speed

When we talk about average speed, we're referring to the total distance traveled divided by the total time taken to travel that distance. In the context of our car problem, calculating the average speed is a bit different since the car isn't traveling at a single speed.

Instead, it's moving at 50 mph initially and then at 60 mph after an hour. To find the average speed for this trip, we need to consider the speeds the car is traveling at over the time period. If the car was traveling at 50 mph for half of the hour and 60 mph for the other half, you would calculate the average speed like this:

• First half-hour: Travels 25 miles at 50 mph
• Second half-hour: Travels 30 miles at 60 mph

The total distance is 25 + 30 = 55 miles and total time is 1 hour.
So, the average speed = Total Distance/Total Time = 55 miles/hour.

Keep in mind that this is a simple average calculation given that we assumed equal time intervals. Real-life scenarios with varying speeds could need weighted averages.

Distance calculation

Distance calculations can determine how far an object has traveled over a certain period of time. In the given problem, we calculate the minimum and maximum distances traveled by the car over an hour by assuming constant speeds.

Here's how it's done:

• Minimum Distance: Assumed the car drives at the slower rate, 50 mph, the entire hour. Thus, it covers \( 50 \) miles in 1 hour.
• Maximum Distance: Assuming it travels at the faster 60 mph for the whole hour, then it covers \( 60 \) miles in 1 hour.

These calculations show that the car's distance is somewhere between these values, given it started slower and ended faster.

This is an example of basic distance calculation in calculus, often applied in constant speed scenarios where actual physical movement may vary.

Rate of change

In calculus, the rate of change is a crucial concept that refers to how a quantity changes over time. In simple terms, it tells us how fast or slow something is moving or changing. When we look at the speed of a car, the speedometer reading gives an instant rate of change, which is the car's speed at a given moment.

In the context of our problem, the car's rate of change in speed varies between 50 mph and 60 mph over one hour. Calculating the average rate of change gives insight into the car's overall performance.

If the car's speed steadily increases or decreases, understanding the rate can indicate something unusual might have happened during the trip, like sudden accelerations or decelerations.

• Initial Rate: 50 mph at start time
• Final Rate: 60 mph at end time

Recognizing rates of change is essential for deeper insights in various scientific and economic fields, as it helps predict future behaviors based on past trends.