A clothing manufacturer produces clothing in five locations in the United States. In a move to vertical integration, the company is planning a new fabric production plant that will supply fabric to all five clothing plants. The clothing plants have been located on a coordinate system as follows: $$\begin{array} { l l }
\text { Location } & ( \mathrm { x } , \mathrm { y } ) \
\hline \mathrm { A } & 7,2 \
\mathrm { B} & 4,7 \
\mathrm { C } & 5,5 \
\mathrm { D } & 6,2 \
\mathrm { E } & 8,4
\end{array}$$ If the amount of fabric shipped to each plant is equal, what is the optimal location for the fabric plant?

Question

A clothing manufacturer produces clothing in five locations in the United States. In a move to vertical integration, the company is planning a new fabric production plant that will supply fabric to all five clothing plants. The clothing plants have been located on a coordinate system as follows: $$\begin{array} { l l } 
\text { Location } & ( \mathrm { x } , \mathrm { y } ) \
\hline \mathrm { A } & 7,2 \
\mathrm { B} & 4,7 \
\mathrm { C } & 5,5 \
\mathrm { D } & 6,2 \
\mathrm { E } & 8,4
\end{array}$$ If the amount of fabric shipped to each plant is equal, what is the optimal location for the fabric plant?

Quizplus · Accepted Answer

The optimal location for the fabric plant is the centroid of the five clothing plants. Using the formula for the centroid, we get:
\begin{align*}
\overline{x} &= \frac{7+4+5+6+8}{5} = 6 \
\overline{y} &= \frac{2+7+5+2+4}{5} = 4 \
\end{align*}
Therefore, the optimal location for the fabric plant is (6,4), which is choice B.

A Clothing Manufacturer Produces Clothing in Five Locations in the United