https://r2.erweima.ai/mathbot/image/1727045843591yplkti3a.jpg

This problem requires determining how long the machines need to work each day after some adjustments in order to complete the production of 120 pieces on time.

### Problem Breakdown:

1. **Initial Plan:**
   - **5 machines**, operating **8 hours per day**.
   - **6 days** to produce 120 pieces.
   - So, in the original plan, each machine was expected to produce the following in one day:
     \[
     \text{Production per day} = \frac{120 \text{ pieces}}{6 \text{ days}} = 20 \text{ pieces per day}
     \]
     Since there are 5 machines working, each machine produces:
     \[
     \frac{20 \text{ pieces}}{5 \text{ machines}} = 4 \text{ pieces per day per machine}.
     \]

2. **Change on Day 4:**
   - At the end of day 4, **3 machines** go into maintenance.
   - They are replaced by **2 new machines**, but these new machines have **half the capacity** of the original ones.
   - The original machines still need to finish the work in the **remaining 2 days**.

### Production Calculation:

- **In the first 4 days**:  
  The original 5 machines work for 8 hours per day for 4 days. Thus:
  \[
  5 \times 4 = 20 \text{ machine-days of work}.
  \]
  Since each machine can produce 4 pieces per day, in 4 days, the total production is:
  \[
  20 \times 4 = 80 \text{ pieces produced}.
  \]
  Therefore, **40 pieces remain** to be produced in the last 2 days.

- **In the last 2 days**:
  - 2 of the original machines remain, and they will work with their full capacity.
  - The 2 replacement machines produce at **half capacity** (i.e., each new machine produces 2 pieces per day).
  
  The production from these 4 machines in one day is:
  \[
  2 \times 4 \text{ (original machines)} + 2 \times 2 \text{ (new machines)} = 8 + 4 = 12 \text{ pieces per day}.
  \]
  In 2 days, these machines would produce:
  \[
  12 \times 2 = 24 \text{ pieces}.
  \]
  So, **16 pieces** would still be missing after the regular 8 hours.

### Extra Hours Needed:

- To produce the remaining 16 pieces, we need to calculate how many more hours the machines must work. The total production rate per hour is:
  \[
  \frac{12 \text{ pieces}}{8 \text{ hours}} = 1.5 \text{ pieces per hour}.
  \]
  To produce the remaining 16 pieces:
  \[
  \frac{16 \text{ pieces}}{1.5 \text{ pieces per hour}} \approx 10.67 \text{ hours}.
  \]
  This is equivalent to 10 hours and 40 minutes.

Since they are already working 8 hours, the total daily time should be:
\[
8 + 1.67 \approx 9 \text{ hours and 40 minutes}.
\]

This isn't quite matching any answer choices exactly, but the closest answer is **B) 9 hours and 8 minutes**.

Would you like further clarification or more details on a specific part of this solution?

Here are 5 related questions:
1. What if the new machines had the same capacity as the original ones?
2. How would the total time change if all machines needed maintenance on day 4?
3. What if the deadline was extended by an extra day?
4. How would the total time per day change if the first 3 days had fewer working hours?
5. What if one more machine was added to replace the ones in maintenance?

**Tip:** When dealing with machine rates, always compare capacity per hour to get the clearest picture of production efficiency.

Para produzir 120 peças de um pedido, estava previsto que 5 máquinas operassem por 6 dias, trabalhando 8 horas por dia. Porém, ao final do quarto dia de trabalho, 3 máquinas tiveram que ser retiradas da produção para manutenção. Outras duas máquinas foram utilizadas para substituí-las, mas elas têm metade da capacidade de produção daquelas que foram para manutenção. Para que o pedido seja entregue no prazo de seis dias, pode-se afirmar que as máquinas (as que se mantiveram produzindo e as que foram usadas para substituir as outras em manutenção) devem continuar operando em um tempo, por dia, de:

Solve a real-world production problem where machine maintenance affects the work rate. Learn how to calculate the necessary time per day to meet the production deadline when machine capacities vary. Suitable for high school students learning work rate concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation