Question

Два робітники, працюючи разом, виконують деяку роботу з а 8 годин. Перший з них, працюючи окремо, може виконати всю роботу на 12 годин скоріше, ніж другий, якщо цей останній працюватиме окремо. За скільки годин перший робітник може виконати всю роботу, працюючи окремо?

Answer 1

Применены : дробно-рациональное уравнение, теорема Виета, формулы зависимости производительности, времени и работы

Answer 2

Problem Analysis

We have two workers who can complete a certain task together in 8 hours. However, if the first worker works alone, they can complete the task 12 hours faster than the second worker working alone. We need to determine how many hours the first worker can complete the task working alone.

Solution

Let's assume that the second worker takes x hours to complete the task working alone. According to the given information, the first worker can complete the task in x - 12 hours working alone.

To find the solution, we can set up the following equation based on the work rates of the two workers:

1/ (x - 12) + 1/x = 1/8

Now, let's solve this equation to find the value of x.

Calculation

To solve the equation, we can multiply both sides by 8x(x - 12) to eliminate the denominators:

8x + 8(x - 12) = x(x - 12)

Simplifying the equation:

8x + 8x - 96 = x^2 - 12x

Combining like terms:

16x - 96 = x^2 - 12x

Rearranging the equation:

x^2 - 28x + 96 = 0

Now, we can solve this quadratic equation to find the value of x.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -28, and c = 96.

Calculating the discriminant:

D = b^2 - 4ac = (-28)^2 - 4(1)(96) = 784 - 384 = 400

Since the discriminant is positive, we have two real solutions for x.

Using the quadratic formula:

x = (-(-28) ± √(400)) / (2(1)) x = (28 ± 20) / 2

Simplifying:

x1 = (28 + 20) / 2 = 48 / 2 = 24 x2 = (28 - 20) / 2 = 8 / 2 = 4

Therefore, the second worker takes 4 hours to complete the task working alone, and the first worker takes 24 hours to complete the task working alone.

Answer

The first worker can complete the entire task working alone in 24 hours.