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Chapter 3: Problem 94

On a 300 -mile auto trip, Lisa averaged 52 mph for the first 100 miles, 65 mph for the second 100 miles, and 58 mph for the last 100 miles. a. How long did the 300 -mile trip take? b. Could you find Lisa's average speed for the 300 -mile trip by calculating \((52+65+58) / 3 ?\) If not, find the correct average speed for the trip.

Short Answer

Expert verified
a) The total time for the 300-mile trip can be calculated as \(1.92 + 100/65 + 100/58\) hours. b) No, the correct average speed for the trip is found by \(300 / (1.92 + 100/65 + 100/58)\) mph, not \((52+65+58)/3\) mph

Step by step solution

01

Calculate time for each leg of the journey

The time for each 100 mile leg can be calculated by dividing the distance (100 miles) by the the average speed for that leg. For example, the time for the first 100 miles would be \(100/52 = 1.92\) hours, about 1 hour and 55 minutes. The other two times are calculated similarly.
02

Calculate the total time for the journey

Now, add up each of the leg times calculated in the previous step to get the total time for the 300 mile journey. That is, \( Total \,\, time= 1.92 + 100 / 65 + 100 / 58 \).
03

Calculate the average speed for the entire trip

The average speed for the entire trip can be found by dividing the total distance travelled by the total time taken. Average speed = \( total \,\, distance/total \,\, time= 300 / \left( 1.92+ (100/65) + (100/58) \right) \) mph.
04

Comparing the calculated average speed to the simple average of the individual speeds

The average speed found in the previous step is not equal to the simple average \((52+65+58)/3\), this is because speed is not a 'linear' quantity. The time each segment of the journey takes is important to consider in calculating average speed, as this problem has illustrated.
05

Final Analysis

This exercise serves as a reminder that not all quantities can be averaged simply by adding together and dividing by the number of values. For speed, each segment's time needs to be taken into account, which is why the correct average speed is calculated by total distance divided by total time instead of simply averaging the speeds.

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