To find the average speed of the cyclist over the entire journey, we need to follow a step-by-step process. Here's how we do it:
1. Determine the times taken for each kilometer:
- The time taken to travel the first kilometer at a speed of 10 km/hr:
[tex]\[
\text{Time for 1st km} = \frac{1 \, \text{km}}{10 \, \text{km/hr}} = 0.1 \, \text{hours}
\][/tex]
- The time taken to travel the second kilometer at a speed of 3 km/hr:
[tex]\[
\text{Time for 2nd km} = \frac{1 \, \text{km}}{3 \, \text{km/hr}} = 0.333\ldots \, \text{hours} \approx 0.333
\][/tex]
- The time taken to travel the third kilometer at a speed of 6 km/hr:
[tex]\[
\text{Time for 3rd km} = \frac{1 \, \text{km}}{6 \, \text{km/hr}} = 0.166\ldots \, \text{hours} \approx 0.167
\][/tex]
2. Calculate the total distance traveled:
- The cyclist travels a total distance of 3 kilometers:
[tex]\[
\text{Total distance} = 1 \, \text{km} + 1 \, \text{km} + 1 \, \text{km} = 3 \, \text{km}
\][/tex]
3. Calculate the total time taken for the journey:
- Adding up the time taken for each segment:
[tex]\[
\text{Total time} = 0.1 \, \text{hours} + 0.333 \, \text{hours} + 0.167 \, \text{hours} = 0.6 \, \text{hours}
\][/tex]
4. Calculate the average speed:
- Average speed is given by the total distance divided by the total time:
[tex]\[
\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{3 \, \text{km}}{0.6 \, \text{hours}} = 5 \, \text{km/hr}
\][/tex]
Therefore, the average speed of the cyclist over the entire journey is [tex]\(5 \, \text{km/hr}\)[/tex].
