A contract is to be completed in 52 days and 125 identical robots were employed, each operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How many additional robots would be required to complete the work on time, if each robot is now operational for 8 hours a day?
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Contract Completion Problem Analysis
This problem involves understanding the relationship between the number of robots, their working hours, the number of days, and the amount of work completed. We will use the concept of "robot-hours" as a unit of work to solve this problem efficiently. The goal is to determine the additional robots required to complete the remaining work on time under new conditions.
Initial Work Phase Details
In the first phase of the contract, we have the following information:
- Initial Robots (M1): 125 identical robots
- Days Worked (D1): 39 days
- Hours per Day (H1): 7 hours/day
- Work Completed (W1): 5/7 of the total contract work
The total work completed in this phase can be calculated in "robot-hours":
\[ \text{Work Done} = M_1 \times D_1 \times H_1 \]
\[ \text{Work Done} = 125 \times 39 \times 7 \]
\[ \text{Work Done} = 34125 \text{ robot-hours} \]
This amount, 34125 robot-hours, represents 5/7 of the total work for the entire contract.
Calculating Total and Remaining Work
Since 34125 robot-hours represent 5/7 of the total work, we can find the total work required for the entire contract:
\[ \text{Total Work} = \text{Work Done} \times \frac{7}{5} \]
\[ \text{Total Work} = 34125 \times \frac{7}{5} \]
\[ \text{Total Work} = (34125 \div 5) \times 7 \]
\[ \text{Total Work} = 6825 \times 7 \]
\[ \text{Total Work} = 47775 \text{ robot-hours} \]
Now, we need to determine the remaining work that needs to be completed:
\[ \text{Remaining Work} = \text{Total Work} - \text{Work Done} \]
\[ \text{Remaining Work} = 47775 - 34125 \]
\[ \text{Remaining Work} = 13650 \text{ robot-hours} \]
Alternatively, the remaining work is \(1 - \frac{5}{7} = \frac{2}{7}\) of the total work:
\[ \text{Remaining Work} = \frac{2}{7} \times 47775 \]
\[ \text{Remaining Work} = 2 \times 6825 \]
\[ \text{Remaining Work} = 13650 \text{ robot-hours} \]
Remaining Time and New Work Conditions
The total contract duration is 52 days, and 39 days have already passed. So, the remaining time to complete the contract is:
- Remaining Days (D2): \(52 - 39 = 13\) days
The new operational hours for each robot are:
- New Hours per Day (H2): 8 hours/day
Calculating Robots Needed for Remaining Work
Let M2 be the total number of robots required to complete the remaining work (13650 robot-hours) within the remaining 13 days, with each robot working 8 hours/day.
\[ M_2 \times D_2 \times H_2 = \text{Remaining Work} \]
\[ M_2 \times 13 \times 8 = 13650 \]
\[ M_2 \times 104 = 13650 \]
To find M2, we divide the remaining work by the total robot-hours one robot can provide in the remaining time:
\[ M_2 = \frac{13650}{104} \]
\[ M_2 = 131.25 \text{ robots} \]
Determining Additional Robots Required
Since it is impossible to employ a fraction of a robot, and we need to ensure the work is completed on time, we must round up the number of robots to the next whole number. Therefore, 132 robots are required to complete the remaining work.
- Total Robots Required (M2): 132 robots
- Initial Robots Employed: 125 robots
The number of additional robots required is the difference between the total robots needed for the second phase and the robots already employed:
\[ \text{Additional Robots} = \text{Total Robots Required} - \text{Initial Robots Employed} \]
\[ \text{Additional Robots} = 132 - 125 \]
\[ \text{Additional Robots} = 7 \text{ robots} \]
Therefore, 7 additional robots are required to complete the work on time.
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