Mathematical modelling problem...
Two towns, A and B, on a straight line 12km apart. I want to put a site on the straight line between the two towns. From the site, travel to A is three times as often as travel to B. I want to position the site so that the amount of travel from the site to the towns is minimized. Assumptions I have given the model are
(i) The towns and the site can be represented by points. (ii) The land is flat. (iii) The mode of transport for the run can travel in straight lines. (iv) The mode of transport is full for each run (v) The run will take three times as often to A as it will to B from the site. (vi) We can sum the journey to both towns to get the total.
All I have is $3x + (12-x)$
I understand that the site needs to very close to A (thanks to comments and answer given already) but for this problem I require an expression in $x$ that I can differentiate, letting $\frac d{dx}=0$ and solving for $x$ and then show that it is a maximized such that the second derivative $>0$
Any solution or help with this would be greatly appreciated
We can assume the site is on the line segment $AB$.
Let $x$ be the distance of the site from $A$, where $0 \le x \le 12$.
The total distance traveled is given by $d=(3n)x + n(12-x)$, where $3n,n$ are the (constant) number of trips to $A,B$, respectively.
Then $d = (2n)x + 12n$, hence for $x \in [0,12]$, the minimum is achieved at $x=0\;$(i.e., place the site at $A$).