Homework SourseA company has retail outlets located at the points 0 0 8 8 a
A company has retail outlets located at the points (0, 0), (8, 8), and (?8, 8) (see figure). Management plans to build a distribution center located such that the sum of the distances S from the center to the outlets is minimum. From the symmetry of the problem it is clear that the distribution center will be located on the y-axis, and therefore S is a function of the single variable y. Using techniques presented in Chapter 3, find the required value of y.
Solution
z={(0-0)^2+(y-0)^2}+{(8-0)^2+(y-8)^2}+{(-8-0)^2+(y-0)^2}
dz/dy=y+.5{64+(y-8)^2}^-.5(2(y-8)+.5{64+y^2}^-.5(2y)=0 multiply both sides by {64+(y-8)^2}.5
0=y{64+(y-8)^2}^.5+y-8+y=y{64+(y-8)^2}^.5+2y-8 then isolate radical
8-2y=y{64+(y-8)^2}^.5 square both sides
64-32y+y^2=y^2{64+y^2-32y+64}
64-32y+y^2-y^2{64+y^2-32y+64}=0 factor
(64-32y+y^2)(1-y^2)=0 set each factor equal to 0
8-2y=0 or y=4
1-y^2=0 or y=1 or -1
when y=0, z=128+64=8(1+2)=19.3
when y=8, z=64+64+128=8(2+2)=27.3
when y=4, z=16+80+80=4(1+25)
when y=1, z=1+113+65=19.6
when y=-1,z=1+145+65>19.6
when y=-2, z=2+164+68>19.6
y=0 is the minimium value
