The Apple Deliverer

Calculus Level 3

An apple deliverer sells his apples to apple shops. From the place where he obtains the apples, there is a shop after every mile. One day he forgets to bring his meal with him and he has to eat the apples for hunger. He has 100 apples and his hunger increases by one apple every mile; that is, he eats \(N\) apples at the \(N^{\text{th}}\) mile.

If every shop gives him $5 more per apple than the previous shop (and first shop will pay $5 per apple) , what shop provides him maximum profit if he wants to sell all of his apples only at one shop? (On each mile, he first eats the apples and then he can sell the remaining apples to the shop.)

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