Your company just won a bid to provide 700000 electronic components to IBM, but

Question

Your company just won a bid to provide 700000 electronic components to IBM, but IBM is in a hurry. For every working day that goes by, the contract decreases in value by $100. It costs $80 to train each worker you use to assemble components and a well-trained worker can produce 5 components per day.

Formula for the total reduction in value of the contract (including paying for workers) after t days.
The reduction in value is 100t+80(700000/(5t)) dollars.

How many workers should you use and how long will you take to fulfill the contract? (Note: in terms of the contract any portion of a day counts as a whole day.)

It will take _____________________________ days to fulfill the contract.

As a result, I should use ______________________ workers.

What must the initial value of the contract be in order for it to be at least break even for your company after the optimal number of days?
It must be worth _____________________________dollars.

Explanation / Answer

Reduction in value of the contract is given by the function R(t)=  100t+80(700000/(5t)) dollars.
To minimise reduction we need to differentiate this function with respect to t and equate it to zero;

Differentiating it with respect to t:
100- 56*1000000/5t^2 = zero;

5t^2= 560000; givign t= 334.66 which means t= 335 (as in terms of the contract any portion of a day counts as a whole day) ; If 7 lakh products are to be produced in 335 days you need number of workers = 700000/ (335* 5)= 417.91 hence 418 workers are to be used.

For the initial value of the contract to be at least break even it means that the total cost incurred due to reduction as the production process was delayed must be equal to the minimum worth of the cost;

Hence minimum worth for break even (MWB) = 100t+80(700000/(5t); Substituting t= 335 gives MWB= 66932.835 $
Hence it must be worth 66933$ at least to be of break even for the manufacturing company

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