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16 technicians can do a task in 5 hours. 10 technicians started the work at 8 am, but one technician is added each hour starting from 11 am. At what time will the task be completed?

Answer :

To solve this problem, we need to understand how the task completion scales with the number of technicians.

Let's start by determining the total work required to complete the task.

  1. Total Work Calculation:
    If 16 technicians can complete the task in 5 hours, the total amount of work can be represented as:

    [tex]ext{Total Work} = 16 ext{ technicians} \times 5 ext{ hours} = 80 ext{ technician-hours}[/tex]

  2. Work Done by 10 Technicians (8 am to 11 am):
    Initial work is done by 10 technicians from 8 am to 11 am:

    [tex]ext{Work from 8 am to 11 am} = 10 ext{ technicians} \times 3 ext{ hours} = 30 ext{ technician-hours}[/tex]

  3. Remaining Work:
    After 3 hours, the remaining work to be completed is:

    [tex]ext{Remaining Work} = 80 ext{ technician-hours} - 30 ext{ technician-hours} = 50 ext{ technician-hours}[/tex]

  4. Adding Technicians from 11 am Onwards:
    The number of technicians changes each hour starting from 11 am. Here's the breakdown:

    • 11 am to 12 pm: 11 technicians (10 original + 1 added)
    • 12 pm to 1 pm: 12 technicians
    • 1 pm to 2 pm: 13 technicians
    • 2 pm to 3 pm: 14 technicians

    Calculate the work done each hour and the cumulative work until the task is finished:

    • 11 am to 12 pm:
      [tex]11 ext{ technicians} \times 1 ext{ hour} = 11 ext{ technician-hours}[/tex]
      Total work now = 30 + 11 = 41 technician-hours

    • 12 pm to 1 pm:
      [tex]12 ext{ technicians} \times 1 ext{ hour} = 12 ext{ technician-hours}[/tex]
      Total work now = 41 + 12 = 53 technician-hours

Since 53 technician-hours exceed the 50 technician-hours needed, the task is completed sometime between 12 pm and 1 pm.

To find exactly when the task is completed after 12 pm:

Partial work required after 12 pm:
[tex]50 ext{ technician-hours} - 41 ext{ technician-hours} = 9 ext{ technician-hours}[/tex]

At the 12 pm - 1 pm interval with 12 technicians available:
[tex]\text{Time to complete} = \frac{9 ext{ technician-hours}}{12 ext{ technicians}} = 0.75 ext{ hours} = 45 ext{ minutes}[/tex]

Thus, the task is completed at 12:45 pm.

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Rewritten by : Barada