Robbie roberts, owner of cupamoe's coffee, is trying to determine where to locate his newest distribution warehouse. robbie would like to locate the warehouse to minimize transportation costs and provide the best overall delivery speed to his three markets. consider the warehouse location decision facing cupamoe's.

1a) We need to find the total population as well as the population in each town.
First, Capital City's population is going to increase by 5% and its population now is 400000. 5%*400000=20000 people and that is the increase in people. Thus, in total there are 420000 people in Capital City. Springfield also has an increase by 50000 people, so its new population is 250000 people.

Now, the total of all the people in the 3 towns is: 420k+250k+170k=840k people. The weight of each city is the fraction of its people over the total population. Hence, we get for the weights:
Capital City: Wc=420/840=1/2
Springfield: Ws=250/840=0.30
Shelbyville: Wb=170/840=0.20
All weights add up to 1, as they should.

Now, in order to calculate the new position, we need to take the sum of the coordinates, by applying the weights. The formula is:
Wc*(Xc, Yc)+Ws*(Xs,Ys)+Wb*(Xb, Yb) where the x and y denote the corresponding coordinates.
Substituting: 0.5*(1,5)+0.3(4.5, 3)+0.2(4,1)=(0.5,2.5)+(1.35, 0.9)+(0.8, 0.2)=(2.65,3.6) by adding coordinate-wise.

1b) We now have to do weights given the sales. Using the same methodology, the total sales are: 800000+150000+150000=1100000$. Hence, calculating the weights we get:
Wc=800000/1100000=0.727
Ws=150000/1100000=0.136
Wb=150000/1100000=0.136
The weights do not add up to 1, due to some rounding errors but that is ok.

Applying the same formula as above: 0.727(1,5)+0.136(4.5,3)+0.136(4,1)=
(0.727, 3.635) + ( 0.612,0.408)+(0.544,0.136)=(1.883, 4.179).
With rounding this yields that our point is (1.88, 4.18). It is important to note that since the weight for Capital City is larger, the end-point is closer to it; hence we have that the y-value is higher than our previous calculation and the x-value is lower.

c) Population is not always a good indicator of sales; it might be that the population does not have a good buying power or regional features might make your product less appealing to a large part of the population (for example Muslim people do not eat pork). On the other hand, sales correlate with products sold since the unit price of the product is relatively constant, so one can estimate the approximate amount of units sold in each town. Hence, the correct response is A.