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Two towns A and B are 100 km apart. A school is to be built for 100 students of Town B and 30 students of Town A. Expenditure on transport is Rs 1.20 per km. If the total expenditure on transport by all 130 students is to be as small as possible, then the school should be built at:

  1. 33 km from Town A
  2. 33 km from town B
  3. Town A 
  4. Town B
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We can write the cost function as :

cost(x)   =   120x + 36(100 - x)

             =   120x  +  3600  - 36x

             =   84x    +  3600

So we can see that it is a strictly increasing function hence x = 0 will give minimum cost i.e. at town B.

If we write w.r.t. town A , we get :

cost(x)   =   36x  + 120(100 - x)

             =   12000 -  84x

So we can see that it is a strictly decreasing function hence x = 100 will give minimum cost i.e. at town B.

So , in either case we get the town B as the place where we get minimum cost.

Hence D) should be correct option.

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