Calc Question - need some help! [Intervals, Slope, and Absolute Values]
Hey, i'm stumped on a question. So am looking from some assistance. Any help appreciated!
Question as follows...
'n' members of my little league club live at different points along a straight road. We want to build a clubhouse on this road, but the problem is where to locate it. We want to minimize the total distance that all members have to walk from their houses to the clubhouse. Where should the clubhouse be located?
It seems like a complicated problem in general (because you seem to have to know about the spacing of n houses), but theres a lovely conceptual solution which involves thinking about rates of change.
Imagine the houses laid out along the x-axis, with the first (westernmost) house at x=0, and think about how the objective function (the sum of the distances of the n houses from the clubhouse) changes as the clubhouse moves east from x=0. Don't try to work out an expression for this function, but think only about the slope of its graph in different subintervals. At what rate does it change as we move from house1 to house2, from house2 to house3, etc. In each case, think of the effect on each member of a small eastwardly movement of the clubhouse.
a) present a general solution with suitable illustrations, for the case of n = 7 members (7 houses). Show that essentially the same solution will work for any odd n.
b) Now do the same case for 8 members - try to formulate a general result.