Question

Из пункта A на прогулку вышел пешеход со скоростью ν км /ч. После того как он отошел от A на 6 км,из A следом за ним выехал велосипедист,скорость которого была на 9 км/ч больше скорости пешехода. Когда велосипедист догнал пешехода,они повернули назад и возвратились вместе A в со скоростью 4км/ч при каком значении ν время прогулки пешехода окажется наименьшим?

Answer 1

Скорость пешехода v км/ч, а вела v+9 км/ч.
Пешеход ушел на 6 км за время t1 = 6/v часов, и тут стартовал вел.
Когда вел догнал пешехода, то они за одно и тоже время t2 прошли:
пешеход - v*t2 км, а вел (v+9)*t2 км = v*t2 + 6 км. Решаем уравнение.
v*t2 + 9*t2 = v*t2 + 6
9*t2 = 6
t2 = 6/9 = 2/3 часа.
Это время не зависит от скорости пешехода v км/ч.
Значит, вел догнал пешехода на расстоянии v*2/3+6 = (2v+18)/3 км от А.
Обратно они поехали со скоростью 4 км/ч и вернулись за
t3 = (2v+18)/(3*4) = (v+9)/6 = v/6 + 9/6 = v/6 + 3/2 часов.
Общее время прогулки пешехода составляет
T = t1 + t2 + t3 = 6/v+2/3+v/6+3/2 = (6/v+v/6) + (4/6+9/6) = (6/v+v/6) + 13/6 ч
Минимальным это значение будет при v = 6 км/ч, тогда 6/v + v/6 = 2.
А общее время прогулки составляет T = 2 + 13/6 = (12+13)/6 = 25/6 часа.
Ответ: v = 6 км/ч, T = 25/6 часа = 4 часа 10 минут.

Answer 2

Calculating the Minimum Time of the Pedestrian's Walk

To calculate the minimum time of the pedestrian's walk, we can follow these steps:

1. Calculate the Time Taken by the Pedestrian to Walk 6 km from Point A - Let's denote the pedestrian's speed as ν km/h. - The time taken by the pedestrian to walk 6 km can be calculated using the formula: time = distance / speed.

2. Calculate the Time Taken by the Cyclist to Catch Up with the Pedestrian - The cyclist's speed is 9 km/h faster than the pedestrian's speed. - We can calculate the time taken by the cyclist to catch up with the pedestrian using the relative speed of the cyclist with respect to the pedestrian.

3. Calculate the Time Taken for the Return Journey at 4 km/h - After the cyclist catches up with the pedestrian, they return together at a speed of 4 km/h.

Let's proceed with the calculations.

Calculation Steps

1. Calculate the Time Taken by the Pedestrian to Walk 6 km from Point A - Let's denote the pedestrian's speed as ν km/h. - The time taken by the pedestrian to walk 6 km is given by the formula: time = distance / speed Substituting the given values: time = 6 km / ν km/h

2. Calculate the Time Taken by the Cyclist to Catch Up with the Pedestrian - The time taken by the cyclist to catch up with the pedestrian can be calculated using the relative speed of the cyclist with respect to the pedestrian. - The relative speed is the difference in their speeds, which is 9 km/h. - The time taken by the cyclist to catch up with the pedestrian is given by the formula: time = distance / relative speed Substituting the given values: time = 6 km / 9 km/h

3. Calculate the Time Taken for the Return Journey at 4 km/h - After the cyclist catches up with the pedestrian, they return together at a speed of 4 km/h. - The time taken for the return journey can be calculated using the formula: time = distance / speed Substituting the given values: time = 6 km / (ν + 4) km/h

Conclusion

To find the value of ν for which the time of the pedestrian's walk is minimized, we need to find the minimum value of the total time taken, which is the sum of the times calculated in the above steps. This involves finding the derivative of the total time with respect to ν and solving for the minimum value.

Let's proceed with the calculations to find the value of ν that minimizes the time of the pedestrian's walk.