8 workers can build a wall 18 m long, 2 m broad and 12 m high in 10 days, working 9 hours a day. Find how many workers will be able to build a wall 32 m long, 3 m broad and 9 m high in 8 days working 6 hours a day?

Options

A. 16
B. 20
C. 30
D. 10

Answer

To solve the problem step by step, we will first calculate the volume of the wall that the first set of workers can build, and then use that information to find out how many workers are needed for the second wall.

### Step 1: Calculate the volume of the first wall
The volume of a wall can be calculated using the formula:
\[ \text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height} \]

For the first wall:
- Length = 18 m
- Breadth = 2 m
- Height = 12 m

So, the volume of the first wall is:
\[ \text{Volume} = 18 \, \text{m} \times 2 \, \text{m} \times 12 \, \text{m} = 432 \, \text{m}^3 \]

### Step 2: Calculate the total work done by the workers
The total work done by the workers can be calculated in terms of worker-hours.

Given:
- Number of workers = 8
- Hours worked per day = 9
- Days worked = 10

Total work done (in worker-hours) is:
\[ \text{Total Work} = \text{Number of Workers} \times \text{Hours per Day} \times \text{Days} \]
\[ \text{Total Work} = 8 \times 9 \times 10 = 720 \, \text{worker-hours} \]

### Step 3: Calculate the work required to build the wall
The work required to build the first wall can be expressed as:
\[ \text{Work Required} = \text{Volume of Wall} \]

We know that 720 worker-hours are required to build 432 m³ of wall. Therefore, the work required per cubic meter is:
\[ \text{Work per m}^3 = \frac{720 \, \text{worker-hours}}{432 \, \text{m}^3} = \frac{720}{432} = \frac{5}{3} \, \text{worker-hours/m}^3 \]

### Step 4: Calculate the volume of the second wall
Now, we need to calculate the volume of the second wall:
- Length = 32 m
- Breadth = 3 m
- Height = 9 m

So, the volume of the second wall is:
\[ \text{Volume} = 32 \, \text{m} \times 3 \, \text{m} \times 9 \, \text{m} = 864 \, \text{m}^3 \]

### Step 5: Calculate the total work required for the second wall
Using the work per cubic meter calculated earlier, the total work required for the second wall is:
\[ \text{Total Work Required} = \text{Volume of Second Wall} \times \text{Work per m}^3 \]
\[ \text{Total Work Required} = 864 \, \text{m}^3 \times \frac{5}{3} \, \text{worker-hours/m}^3 = 1440 \, \text{worker-hours} \]

### Step 6: Calculate the number of workers needed
Now we need to find out how many workers are required to complete this work in the given time.

Given:
- Days = 8
- Hours per Day = 6

Total available worker-hours with \( x \) workers is:
\[ \text{Total Worker-Hours} = x \times 6 \times 8 = 48x \]

Setting this equal to the total work required:
\[ 48x = 1440 \]

Now, solving for \( x \):
\[ x = \frac{1440}{48} = 30 \]

### Final Answer
Thus, **30 workers** will be able to build the second wall in the given time.

---