Question

Бассейн, объем которого 425 м(кубических), можно наполнить за 17ч., если одновременно открыть оба крана. Однако наполняя бассейн так, что первый кран был открыт на 5ч. дольше, чем второй. Если первый кран открыть на столько часов, сколько был открыт второй, а второй - на столько, сколько был открыт первый, то через первый кран поступит вдвое меньше воды, чем через второй.Сколько времени был открыт второй кран?

Answer 1

За 1 час через 1 кран поступит x л воды, а через 2 кран y л воды
17(x + y) = 425
x + y = 25
y = 25 - x
Первый кран был открыт а часов, а второй кран а - 5 часов
ax + (a - 5)(25 - x) = 425
Если первый кран открыть на а - 5 часов, а второй на а часов, то через первый зальется в 2 раза меньше.
x(a - 5)*2 = (25 - x)*a
Получили систему из 2 уравнений с 2 неизвестными
{ ax + (a - 5)(25 - x) = 425
{ 2x(a - 5) = (25 - x)*a
Раскрываем скобки
{ ax + 25a - 125 - ax + 5x = 425
{ 2ax - 10x - 25a + ax = 0
Приводим подобные
{ 25a + 5x = 550
{ 3ax - 10x - 25a = 0
Делим 1 уравнение на 5 и выражаем х через а
{ 5a + x = 110, x = 110 - 5a
{ 3ax - 10x - 25a = 0
3a(110 - 5a) - 10(110 - 5a) - 25a = 0
Делим уравнение на 5 и раскрываем скобки
3a*22 - 3a^2 - 220 + 10a - 5a = 0
Меняем знак и приводим подобные
3a^2 - 71a + 220 = 0
D = 71^2 - 4*3*220 = 5041 - 2640 = 2401 = 49^2
a1 = (71 + 49)/6 = 120/6 = 20
a2 = (71 - 49)/6 = 22/6 < 5 - не может быть, потому что в уравнении было положительное число а - 5.
Значит, а = 20, а второй кран был открыт а - 5 = 20 - 5 = 15 часов.
Производительность кранов
x = 110 - 5a = 110 - 5*20 = 10 л/час, y = 25 - x = 25 - 10 = 15 л/ч

Answer 2

Problem Analysis

We are given a pool with a volume of 425 cubic meters that can be filled in 17 hours if both taps are opened simultaneously. However, if the first tap is opened for 5 hours longer than the second tap, the first tap will deliver half the amount of water compared to the second tap. We need to determine how long the second tap was open.

Solution

Let's assume that the second tap was open for x hours. According to the problem, the first tap was open for x + 5 hours.

To solve this problem, we can use the concept of work. The work done by each tap is equal to the product of the rate at which water flows and the time the tap is open. Since the volume of the pool is constant, the total work done by both taps should be equal to the volume of the pool.

Let's denote the rate at which water flows through the first tap as r1 and the rate at which water flows through the second tap as r2.

According to the problem, if the first tap is open for x + 5 hours and the second tap is open for x hours, the first tap delivers half the amount of water compared to the second tap. Mathematically, we can express this as:

(r1 * (x + 5)) = 0.5 * (r2 * x)

We also know that if both taps are open simultaneously, the pool can be filled in 17 hours. Mathematically, we can express this as:

(r1 + r2) * 17 = 425

We now have a system of two equations with two variables (r1 and r2). We can solve this system of equations to find the values of r1 and r2, and then determine the value of x.

Let's solve the system of equations:

Equation 1: (r1 * (x + 5)) = 0.5 * (r2 * x) Equation 2: (r1 + r2) * 17 = 425

From Equation 2, we can express r1 in terms of r2:

r1 = (425 - (r2 * 17)) / 17

Substituting this value of r1 into Equation 1, we get:

((425 - (r2 * 17)) / 17) * (x + 5) = 0.5 * (r2 * x)

Simplifying this equation, we get:

425 - (r2 * 17) = 0.5 * r2 * x + 0.5 * r2 * 5

425 - (r2 * 17) = 0.5 * r2 * (x + 5)

Dividing both sides of the equation by 0.5 * r2, we get:

850 - (r2 * 34) = x + 5

x = 850 - (r2 * 34) - 5 x = 845 - (r2 * 34)

Now, we can substitute the value of x into Equation 2 to find the value of r2:

(r1 + r2) * 17 = 425

((425 - (r2 * 17)) / 17) + r2 = 425 / 17

425 - (r2 * 17) + 17 * r2 = 425

17 * r2 - r2 * 17 = 425 - 425

0 = 0

Since the equation simplifies to 0 = 0, it means that r2 can have any value. Therefore, there are infinitely many possible values for the time the second tap was open.

In conclusion, we cannot determine the exact time the second tap was open based on the given information.