A cyclist covers first five kilometers at an average speed of 10 k.m. per hour, another three kilometers at 8 k.m. per hour and the last two kilometers at 5 k.m. per hour. Then, the average speed of the cyclist during the whole journey, is
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A common mistake is to simply average the given speeds (10, 8, and 5). This is incorrect because the cyclist spends different amounts of time traveling at each speed. Always use the formula: Total Distance / Total Time.
Step 1: Concept of Average Speed:
Average speed is total distance divided by total time, not the average of individual speeds. Step 2: Calculation Steps:
1. Find the total distance.
2. Calculate the time for each segment: Time = Distance / Speed.
3. Sum the times to get the total time.
4. Calculate average speed: Average Speed = Total Distance / Total Time. Step 3: Detailed Solution: 1. Total Distance:
Total distance is the sum of each segment's distance:
\[ \text{Total Distance} = 5 \text{ km} + 3 \text{ km} + 2 \text{ km} = 10 \text{ km} \] 2. Segment Times:
Segment 1 (\(t_1\)): Distance = 5 km, Speed = 10 km/hr.
\[ t_1 = \frac{5 \text{ km}}{10 \text{ km/hr}} = 0.5 \text{ hours} \]
Segment 2 (\(t_2\)): Distance = 3 km, Speed = 8 km/hr.
\[ t_2 = \frac{3 \text{ km}}{8 \text{ km/hr}} = 0.375 \text{ hours} \]
Segment 3 (\(t_3\)): Distance = 2 km, Speed = 5 km/hr.
\[ t_3 = \frac{2 \text{ km}}{5 \text{ km/hr}} = 0.4 \text{ hours} \] 3. Total Time:
Total time is the sum of the segment times:
\[ \text{Total Time} = t_1 + t_2 + t_3 = 0.5 + 0.375 + 0.4 = 1.275 \text{ hours} \] 4. Average Speed:
Calculating the average speed:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{10 \text{ km}}{1.275 \text{ hours}} \approx 7.843 \text{ km/hr} \] Step 4: Answer:
The cyclist's average speed for the entire journey is approximately 7.84 km/hr.