Answer :
Sure! Let's address each part of the question systematically, step by step.
### Part (a): Determining the number of additional men needed if working 8 hours per day
Firstly, we know from the question the following information:
- Total length of the road to be laid: 3000 meters
- Total time to lay the road: 30 days
- Initial number of men employed: 50 men
- Initial working hours per day: 8 hours per day
- Length of the road completed after 20 days: 1200 meters
#### Step 1: Calculate the remaining road length
First, we determine the amount of road left to be constructed after 20 days:
[tex]\[ \text{Remaining road length} = \text{Total road length} - \text{Road completed} = 3000 \text{ m} - 1200 \text{ m} = 1800 \text{ m} \][/tex]
#### Step 2: Calculate the remaining days
Next, we calculate the days left to complete the road:
[tex]\[ \text{Remaining days} = \text{Total days} - \text{Days passed} = 30 \text{ days} - 20 \text{ days} = 10 \text{ days} \][/tex]
#### Step 3: Calculate the initial work rate (road laying rate)
To understand the pace at which the initial team worked, we calculate the road laying rate:
[tex]\[ \text{Total man-hours in the first part} = \text{Initial men} \times \text{Initial hours per day} \times \text{Days passed} = 50 \text{ men} \times 8 \text{ hours/day} \times 20 \text{ days} = 8000 \text{ man-hours} \][/tex]
[tex]\[ \text{Road laying rate} = \frac{\text{Total road laid}}{\text{Total man-hours in first part}} = \frac{1200 \text{ m}}{8000 \text{ man-hours}} = 0.15 \text{ meters/man-hour} \][/tex]
#### Step 4: Calculate required man-hours to finish the remaining road
Now, calculate the required man-hours to complete the rest of the road:
[tex]\[ \text{Required man-hours for remaining road} = \frac{\text{Remaining road length}}{\text{Road laying rate}} = \frac{1800 \text{ m}}{0.15 \text{ meters/man-hour}} = 12000 \text{ man-hours} \][/tex]
#### Step 5: Calculate the number of men needed, working 8 hours per day
Finally, determine the number of men required to finish the road in the remaining 10 days, working 8 hours per day:
[tex]\[ \text{Required men} = \frac{\text{Required man-hours}}{\text{Remaining days} \times \text{Hours per day}} = \frac{12000 \text{ man-hours}}{10 \text{ days} \times 8 \text{ hours/day}} = 150 \text{ men} \][/tex]
Since initially there were 50 men, the additional number of men needed is:
[tex]\[ \text{Additional men needed} = 150 - 50 = 100 \text{ men} \][/tex]
### Part (b): Determining the number of men needed if working 10 hours per day
#### Step 1: Calculate the number of men needed, working 10 hours per day
Using the previously calculated required man-hours (12000 man-hours) and the remaining days (10 days), we determine how many men are needed if they now work 10 hours a day:
[tex]\[ \text{Required men} = \frac{\text{Required man-hours}}{\text{Remaining days} \times \text{New hours per day}} = \frac{12000 \text{ man-hours}}{10 \text{ days} \times 10 \text{ hours/day}} = 120 \text{ men} \][/tex]
Since initially there were 50 men, the additional number of men needed is:
[tex]\[ \text{Additional men needed} = 120 - 50 = 70 \text{ men} \][/tex]
Summary:
(a) The contractor needs to employ 100 additional men if working 8 hours per day.
(b) The contractor needs to employ 70 additional men if working 10 hours per day.
### Part (a): Determining the number of additional men needed if working 8 hours per day
Firstly, we know from the question the following information:
- Total length of the road to be laid: 3000 meters
- Total time to lay the road: 30 days
- Initial number of men employed: 50 men
- Initial working hours per day: 8 hours per day
- Length of the road completed after 20 days: 1200 meters
#### Step 1: Calculate the remaining road length
First, we determine the amount of road left to be constructed after 20 days:
[tex]\[ \text{Remaining road length} = \text{Total road length} - \text{Road completed} = 3000 \text{ m} - 1200 \text{ m} = 1800 \text{ m} \][/tex]
#### Step 2: Calculate the remaining days
Next, we calculate the days left to complete the road:
[tex]\[ \text{Remaining days} = \text{Total days} - \text{Days passed} = 30 \text{ days} - 20 \text{ days} = 10 \text{ days} \][/tex]
#### Step 3: Calculate the initial work rate (road laying rate)
To understand the pace at which the initial team worked, we calculate the road laying rate:
[tex]\[ \text{Total man-hours in the first part} = \text{Initial men} \times \text{Initial hours per day} \times \text{Days passed} = 50 \text{ men} \times 8 \text{ hours/day} \times 20 \text{ days} = 8000 \text{ man-hours} \][/tex]
[tex]\[ \text{Road laying rate} = \frac{\text{Total road laid}}{\text{Total man-hours in first part}} = \frac{1200 \text{ m}}{8000 \text{ man-hours}} = 0.15 \text{ meters/man-hour} \][/tex]
#### Step 4: Calculate required man-hours to finish the remaining road
Now, calculate the required man-hours to complete the rest of the road:
[tex]\[ \text{Required man-hours for remaining road} = \frac{\text{Remaining road length}}{\text{Road laying rate}} = \frac{1800 \text{ m}}{0.15 \text{ meters/man-hour}} = 12000 \text{ man-hours} \][/tex]
#### Step 5: Calculate the number of men needed, working 8 hours per day
Finally, determine the number of men required to finish the road in the remaining 10 days, working 8 hours per day:
[tex]\[ \text{Required men} = \frac{\text{Required man-hours}}{\text{Remaining days} \times \text{Hours per day}} = \frac{12000 \text{ man-hours}}{10 \text{ days} \times 8 \text{ hours/day}} = 150 \text{ men} \][/tex]
Since initially there were 50 men, the additional number of men needed is:
[tex]\[ \text{Additional men needed} = 150 - 50 = 100 \text{ men} \][/tex]
### Part (b): Determining the number of men needed if working 10 hours per day
#### Step 1: Calculate the number of men needed, working 10 hours per day
Using the previously calculated required man-hours (12000 man-hours) and the remaining days (10 days), we determine how many men are needed if they now work 10 hours a day:
[tex]\[ \text{Required men} = \frac{\text{Required man-hours}}{\text{Remaining days} \times \text{New hours per day}} = \frac{12000 \text{ man-hours}}{10 \text{ days} \times 10 \text{ hours/day}} = 120 \text{ men} \][/tex]
Since initially there were 50 men, the additional number of men needed is:
[tex]\[ \text{Additional men needed} = 120 - 50 = 70 \text{ men} \][/tex]
Summary:
(a) The contractor needs to employ 100 additional men if working 8 hours per day.
(b) The contractor needs to employ 70 additional men if working 10 hours per day.
