Question

На одному верстаті партію деталей обробляють на три дні довше, ніж на другому. Скільки днів треба для обробки цієї партії деталей верстатом з більшою продуктивністю, якщо відомо, що при сумісній праці на цих верстатах у 3 рази більша партія деталей була оброблена за 20 днів.

Answer 1

Пусть на одном станке партию обработают за x дней, тогда на другом за (x+3) дней. Производительность одного (бОльшая) $\frac1x$, другого $\frac1{x+3}$. За 20 дней выполнят втрое больший заказ:
$\left(\frac1x+\frac1{x+3}\right)\cdot20=3\\\frac{x+3+x}{x^2+3x}=\frac3{20}\\40x+60=3x^2+9x\\3x^2-31x-60=0\\D=961+4\cdot3\cdot60=1681=(41)^2\\x_{1,2}=\frac{31\pm41}6\\x_1=12\\x_2=-2\frac23\;-\;He\;nogx.$

Ответ: станок с большей производительностью обработает партию за 12 дней.

Answer 2

Calculating Processing Time for Parts on Different Machines

To calculate the processing time for a batch of parts on two different machines, we can use the information provided. Let's denote the time taken by the first machine as T1 and the time taken by the second machine as T2.

Given: - The batch of parts takes three days longer to process on the first machine compared to the second machine. - When both machines work together, a batch three times larger was processed in 20 days.

We can set up the following equations based on the given information:

1. T1 = T2 + 3 (The batch of parts takes three days longer to process on the first machine compared to the second machine.) 2. 20(T1 + T2) = 3T1 + 3T2 (When both machines work together, a batch three times larger was processed in 20 days.)

Let's solve these equations to find the processing time for the batch of parts on the machine with higher productivity.

Solving the Equations

First, let's express T1 in terms of T2 using the equation T1 = T2 + 3:

T1 = T2 + 3

Substitute T1 in the second equation:

20((T2 + 3) + T2) = 3(T2 + 3) + 3T2

Now, solve for T2:

20(2T2 + 3) = 3T2 + 9 + 3T2

40T2 + 60 = 6T2 + 9

40T2 - 6T2 = 9 - 60

34T2 = -51

T2 = -51 / 34

However, the solution for T2 is negative, which doesn't make sense in this context. It's possible that there might be a mistake in the given information or the setup of the equations. If you have additional information or if there's a different approach you'd like to take, please feel free to share it!