An athlete prepares for the Comrades. He jogs out to a certain point and then turns back home. On the first day he turns after 6m.on the second day he turns after 10km. On the third day he turns after 14km.
1.after how many kilometres does he turn on the 7th day?
2.On which day did he jog 38km in total?
3how many kilometres has he jogged in total when he reached his house on the 10th day?
4.is it possible that this way of preparing for a race can carry on like this for an indefinite period of time?give a reason for your answer.
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1. To determine the pattern, let's find the difference in distance turned between consecutive days:
- Day 2: 10km - 6km = 4km
- Day 3: 14km - 10km = 4km
We can observe that there is an increase of 4 km in the distance turned each day. Therefore, on the 7th day, the athlete would turn after:
- Day 7: 6km + (4km * (7-1)) = 6km + (4km * 6) = 30km
2. To find the day when the athlete jogged 38km in total, we need to find the day where the total distance turned exceeds 38km.
Let's calculate the total distance turned on each day:
- Day 1: 6km
- Day 2: 6km + 4km = 10km
- Day 3: 10km + 4km = 14km
- Day 4: 14km + 4km = 18km
- Day 5: 18km + 4km = 22km
- Day 6: 22km + 4km = 26km
- Day 7: 26km + 4km = 30km
- Day 8: 30km + 4km = 34km
- Day 9: 34km + 4km = 38km
Therefore, the athlete jogged a total of 38km on the 9th day.
3. To find the total distance jogged by the athlete when they reached their house on the 10th day, we need to sum up the distances turned on each day from Day 1 to Day 10:
- Day 1: 6km
- Day 2: 6km + 4km = 10km
- Day 3: 10km + 4km = 14km
- Day 4: 14km + 4km = 18km
- Day 5: 18km + 4km = 22km
- Day 6: 22km + 4km = 26km
- Day 7: 26km + 4km = 30km
- Day 8: 30km + 4km = 34km
- Day 9: 34km + 4km = 38km
- Day 10: 38km + 4km = 42km
Therefore, the athlete has jogged a total of 42km when they reached their house on the 10th day.
4. No, this way of preparing for a race cannot carry on like this indefinitely. The pattern shows that the athlete is increasing the distance turned by 4km each day. Eventually, the athlete would reach a point where the distance turned becomes unachievable or impractical due to physical limitations. For instance, it would be difficult for an athlete to turn after 50km, 54km, 58km, etc., on consecutive days. At some point, the athlete's physical capabilities would prevent them from further increasing the distance turned.
To find the answers to these questions, we need to analyze the pattern and calculate the distances for each day.
1. To determine how many kilometers the athlete turns on the 7th day, we need to observe the pattern. The athlete is incrementing the distance by 4km each day (6km, 10km, 14km). Since each day the athlete increases the distance by 4km, on the 7th day, the athlete will turn back after 6km + (4km * 6) = 30km.
2. To find the day on which the athlete jogged a total of 38km, we need to calculate the cumulative distances for each day until it reaches or exceeds 38km. We can observe a pattern that the athlete is increasing the distance jogged by 4km each day. So, we can calculate the cumulative distances:
- On day 1: 6km
- On day 2: 6km + 10km = 16km
- On day 3: 16km + 14km = 30km
- On day 4: 30km + 18km = 48km
Since the total distance jogged on day 4 is 48km, which exceeds 38km, the athlete jogged a total of 38km on day 4.
3. To calculate the total distance jogged when the athlete reaches home on the 10th day, we can continue the pattern. The athlete incrementally increases the distance jogged by 4km each day. So, the cumulative distances will be:
- On day 1: 6km
- On day 2: 6km + 10km = 16km
- On day 3: 16km + 14km = 30km
- On day 4: 30km + 18km = 48km
- On day 5: 48km + 22km = 70km
- On day 6: 70km + 26km = 96km
- On day 7: 96km + 30km = 126km
- On day 8: 126km + 34km = 160km
- On day 9: 160km + 38km = 198km
- On day 10: 198km + 42km = 240km
Therefore, when the athlete reaches home on the 10th day, the athlete would have jogged a total of 240km.
4. It is possible for the athlete to continue preparing in this manner indefinitely. The pattern shows that the athlete incrementally increases the distance jogged by 4km each day. As long as the athlete can physically handle the increasing distance, this method can continue. However, at some point, the athlete may reach the limit of their physical capability and need to modify their training approach.
By analyzing the pattern and using simple mathematical calculations, we can answer the questions and understand the logic behind them.