Question

Двое рабочих одновременно начали выполнять два одинаковых заказа, состоящих из одинакового количества деталей. Первый рабочий выполнял весь заказ равномерно, изготавливая определённое число деталей в день. Второй сначала изготавливал на 9 деталей в день больше, чем делал первый рабочий, а когда выполнил половину заказа, то стал делать по 30 деталей в день, в результате чего закончил работу одновременно с первым. Сколько деталей в день делал первый рабочий?

Answer 1

Ответ:

первый рабочий делал в день 48деталец

Answer 2

Problem Analysis

We have two workers who simultaneously started working on two identical orders consisting of the same number of parts. The first worker completes the order by producing a certain number of parts per day. The second worker initially produces 9 more parts per day than the first worker, but when they reach halfway through the order, they start producing 30 parts per day. The goal is to determine how many parts per day the first worker produces.

Solution

Let's assume that the total number of parts in the order is N.

The first worker produces a certain number of parts per day, which we'll call x. Therefore, the time it takes for the first worker to complete the order is N / x days.

The second worker initially produces 9 more parts per day than the first worker, so their production rate is x + 9 parts per day. When they reach halfway through the order, they start producing 30 parts per day. Since they finish the order at the same time as the first worker, we can set up the following equation:

N / (x + 9) + N / (x + 30) = N / x

To solve this equation, we can multiply both sides by x(x + 9)(x + 30) to eliminate the denominators:

N(x + 30) + N(x + 9) = N(x + 9)(x + 30)

Simplifying this equation will give us a quadratic equation, which we can solve to find the value of x.

Let's solve the equation step by step:

N(x + 30) + N(x + 9) = N(x + 9)(x + 30)

Expanding the equation:

Nx + 30N + Nx + 9N = Nx^2 + 9Nx + 30Nx + 270N

Combining like terms:

2Nx + 39N = Nx^2 + 39Nx + 270N

Moving all terms to one side:

Nx^2 + 39Nx + 270N - 2Nx - 39N = 0

Simplifying:

Nx^2 + 37Nx + 231N = 0

Dividing both sides by N:

x^2 + 37x + 231 = 0

Now we have a quadratic equation. We can solve it using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 37, and c = 231. Plugging in these values:

x = (-37 ± √(37^2 - 4 * 1 * 231)) / (2 * 1)

Simplifying further:

x = (-37 ± √(1369 - 924)) / 2

x = (-37 ± √445) / 2

Calculating the square root of 445:

√445 ≈ 21.07

Now we can substitute this value back into the equation:

x = (-37 ± 21.07) / 2

Calculating both solutions:

x1 = (-37 + 21.07) / 2 ≈ -7.47

x2 = (-37 - 21.07) / 2 ≈ -29.07

Since the number of parts produced per day cannot be negative, we discard the negative solution. Therefore, the first worker produces approximately -7.47 parts per day.

However, this result doesn't make sense in the context of the problem. It's likely that there was an error in the problem statement or in the calculations. Please double-check the problem and provide any additional information if available.