Question

Двое трактористов, работая вместе, могут вспахать поле за 6ч. За сколько часов может вспахать это поле каждый тракторист, работая самостоятельно, если одному из них для того, чтобы вспахать 2/3 поля, необходимо на 4ч больше, чем другому, чтобы вспахать 0,2 поля?

Answer 1

Пусть х ч - время 1-го тракториста, у ч - время 2-го тракториста. 1/х пашет за 1 час 1-й тракторист, 1/у - пашет за 1 час 2-й тракторист. 1/х+1/у - пашут вместе за 1 час 1/(1/х+1/у) = 6 ч - вспашут всё поле, работая вместе. (1 уравнение) 2/5 : 1/х час. - время 1-го, за которое он вспашет 2/5 поля. Это на 4 ч больше, чем 1/5 : 1/у час - время 2-го, за которое он вспашет 1/5 поля. Составляем 2-е уравнение 2х/5 - у/5 = 4. Упрощаем каждое и получаем систему уравнений: ху=6(х+у) и 2х-у=20 Из второго у=2х-20, подставляем в первое х(2х-20)=6(х+2х-20) 2х*х-20х-18х+120=0 2х*х-38х+120=0 х*х-19х+60=0 х1=4, х2=15. Подставляе и находим у: у1=-12, у2=10. Первая пара - посторонние корни, т.к. у1 должно быть больше 0. ОТВЕТ: время 1-го тракториста - 15 часов время 2-го тракториста - 10 часов ПРОВЕРКА: 1/(1/15+1/10)=150/25=6 ч, 2/5:1/15=30/5=6 ч., 1/5:1/10=10/5=2 ч. 6>2 на 4 часа.

Answer 2

Calculating Individual Plowing Times

To calculate the individual plowing times for each tractor driver, we can use the information provided.

Let's denote: - The time it takes for the first tractor driver to plow the entire field as x hours. - The time it takes for the second tractor driver to plow the entire field as y hours.

We are given that when working together, they can plow the field in 6 hours. This can be expressed as the equation: 1/x + 1/y = 1/6.

Additionally, we know that one of the drivers takes 4 hours longer to plow 2/3 of the field compared to the other driver plowing 0.2 of the field.

Let's denote: - The time it takes for the first tractor driver to plow 2/3 of the field as a hours. - The time it takes for the second tractor driver to plow 0.2 of the field as b hours.

We can express the given relationship as: a = b + 4.

Solving the Equations

We can solve the system of equations to find the individual plowing times for each tractor driver.

First, we can solve the equation 1/x + 1/y = 1/6 for y: 1/y = 1/6 - 1/x 1/y = (x - 6) / 6x y = 6x / (x - 6)

Next, we can substitute the value of y into the equation a = b + 4: a = (6x / (x - 6)) - 4 + b

We also know that the time it takes for the first tractor driver to plow 2/3 of the field is 4 hours longer than the time it takes for the second tractor driver to plow 0.2 of the field: a = b + 4

By equating the two expressions for a, we can solve for b in terms of x: (6x / (x - 6)) - 4 + b = b + 4 6x / (x - 6) - 4 = 8 6x = 8(x - 6) 6x = 8x - 48 48 = 2x x = 24

Now that we have found the value of x, we can substitute it back to find the value of y: y = 6x / (x - 6) y = 6*24 / (24 - 6) y = 144 / 18 y = 8

Conclusion

Therefore, the first tractor driver can plow the field in 24 hours, and the second tractor driver can plow the field in 8 hours when working individually.