The Apple Deliverer

The number of apples eaten per mile is equal to the number of miles traveled. So, the total number of apples eaten at any given mile would be the sum of all the apples eaten before as well as those eaten at that mile. e.g. $1+2+3+4=10$ apples eaten after 4 miles This follows the triangular numbers series, which is represented as $\frac { { n }^{ 2 }+n }{ 2 }$ . The price at which the shop buys the apples is equal to the distance traveled times 5, $5n$ . From these two equations we understand that the total revenue therefore will be equal to

$5n\left( 100-\left( \frac { { n }^{ 2 }+n }{ 2 } \right) \right)$ The vertex of this expression is $\boxed { \quad (8,2560)\quad }$ Telling us that after eight miles the revenue will be the highest it can be, so to maximize his revenue he should sell his apples to the eighth shop.

Let us first write the money he makes on the $n^{\text{th}}$ town per apple .

He makes $\$ 5$ on the first down, $\$ 5 + \$ 5 = \$ 10$ on the second town, $\$ 5 + \$ 5 + \$ 5 = \$ 15$ on the third town. So that is simply $5n$ per apple on the $n^{\text{th}}$ town.

Now, how many apples does he have to sell at the $n^{\text{th}}$ town? Well, if he sold all the apples on the $3^{\text{rd}}$ town, for example, he would have $100 - \left( 1 + 2 + 3 \right)$ apples remaning. So on the $n^{\text{th}}$ town, he has $100 - \left( 1 + 2 + \dots + n \right)$ apples remaning, which is simply $100 - \left( \dfrac{n(n+1)}{2} \right)$ .

Thus, the net amount he makes by selling apples at the $n^{\text{th}}$ town is, as a function of $n$ ,

$\Rightarrow f(n) = 5n \left( 100 - \left( \dfrac{n(n+1)}{2} \right) \right)$

We need to find the maximum value of $f(n)$ . Thus, by taking the derivative, and finding $f'(n)$ and equating it to $0$ , we find the maximum of $f(n)$ at $n = - \dfrac{1}{3} + \dfrac{\sqrt{601}}{3} \approx 7.83$ . Thus, he would make a maximum profit on the $8^{\text{th}}$ town, and thus, our answer is $\boxed{8}$ .

solve 3x^2+2x=200 using given conditions and u will get x = 7.83 approx.. so it may be 7 r 8 substitute those 7 and 8 u will find 8 will be required shop