Question

The table shows the data of running a machine for five years. The original machine cost is Rupees 70,000. In order to minimize the average total cost per year for running the machine, the machine should be replaced after ............. years. (Answer in integer) % Table \[ \begin{array}{|c|c|c|c|c|c|} \hline & {1{st} year} & {2{nd} year} & {3{rd} year} & {4{th} year} & {5{th} year}
\hline {Resale value (Rupees)} & 40000 & 30000 & 25000 & 22000 & 20000
\hline {Maintenance cost (Rupees)} & 19100 & 20300 & 23500 & 30500 & 40000
\hline \end{array} \]

Hint:

To minimize the average total cost per year, compare the average costs for each year and select the year with the lowest value.

Answer 1

Correct Answer: 4

We are given the following data: - Original machine cost = 70,000 Rupees. - Resale values and maintenance costs for each year. The total cost for each year is calculated by the formula: \[ {Total Cost} = {Original Machine Cost} + \sum {Maintenance Costs for each year} - {Resale Value at the end of year } n \] For each year, we calculate the total cost and then determine the average total cost per year. Year 1: \[ TC_1 = 70000 + 19100 - 40000 = 49100 \] \[ {Average Cost}_1 = \frac{49100}{1} = 49100 \] Year 2: \[ TC_2 = 70000 + 19100 + 20300 - 30000 = 89400 \] \[ {Average Cost}_2 = \frac{89400}{2} = 44700 \] Year 3: \[ TC_3 = 70000 + 19100 + 20300 + 23500 - 25000 = 107900 \] \[ {Average Cost}_3 = \frac{107900}{3} = 35966.67 \] Year 4: \[ TC_4 = 70000 + 19100 + 20300 + 23500 + 30500 - 22000 = 141400 \] \[ {Average Cost}_4 = \frac{141400}{4} = 35350 \] Year 5: \[ TC_5 = 70000 + 19100 + 20300 + 23500 + 30500 + 40000 - 20000 = 213400 \] \[ {Average Cost}_5 = \frac{213400}{5} = 42680 \] The average cost is minimized after 4 years. Therefore, the machine should be replaced after \( \boxed{4} \) years.