The ill Worker

Algebra Level 5

A worker is given a job, but as he was going to start the job he got ill due to flu. If he might not have been got ill, he could have done the job in 50 days. His health conditions are going worse constantly and his work per unit time capacity is decreasing constantly (linearly) with respect to the amount of work done and at the end of work he has half the work per unit time capacity left as compared to that of at the time of starting of work. Find out the time taken by worker (in days) to complete the job. Give your answer in nearest integer.

The answer is 69.

1 solution

Arashdeep Singh
Mar 3, 2014

Let us assume the total work to be 'W'. Then the worker's starting capacity to do work is W/50 and its working capacity at end of job is (W/50)/2 = W/100, this varies uniformly with the amount of work done. Now relation of work capacity to work done is (for this case) : $\frac { W }{ 50 } -\frac { w }{ 100 }$ where w is variable. Now for finding total time required we have to intigrate it as follows:

$\int _{ 0 }^{ W }{ \frac { dw }{ \left( \cfrac { W }{ 50 } -\cfrac { w }{ 100 } \right) } }$

= ${ \left[ \ln { \left( \cfrac { W }{ 50 } -\cfrac { w }{ 100 } \right) \times \cfrac { 1 }{ { 1 }/{ -100 } } } \right] }_{ 0 }^{ W }$

= $-100\left[ \ln { \left( \cfrac { W }{ 50 } -\cfrac { W }{ 100 } \right) } -\ln { \left( \cfrac { W }{ 50 } -0 \right) } \right]$

= $100\left[ \ln { \left( \frac { { W }/{ 50 } }{ { W }/{ 100 } } \right) } \right]$

= $100\ln { \left( 2 \right) }$

= 69.315 ~ 69 days which is the answer.

I know this question involves much calculus than algebra and sorry to put it in wrong category.

This question is purely my own creation.

The ill Worker

The answer is 69.

1 solution

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