Answer to Question #157343 in Statistics and Probability for izzah XV

Question

East Texas Seasonings is preparing to build one processing center to serve its four sources of seasonings. The four source locations are at coordinates shown below. Also, the volume from each source is provided. What is the center of gravity?

X-coordinate

Y-coordinate

Volume

Athens, Texas

30

30

150

Beaumont, Texas

20

10

350

Carthage, Texas

10

70

100

Denton, Texas

50

50

400

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & x & y & Volume \\ \hline Athens, Texas & 30 & 30 & 150 \\ \hdashline Beaumont, Texas & 20 &10 & 350 \\ \hdashline Carthage, Texas & 10 &70 & 100 \\ \hdashline Denton, Texas & 50 &50 & 400 \\ \end{array}$$

The coordinates of the center of gravity can be identified by

$$x_c =\dfrac{\sum x_iV_i}{\sum V_i}$$

$$y_c =\dfrac{\sum y_iV_i}{\sum V_i}$$

$$\sum x_iV_i=30(150)+20(350)+10(100)+50(400)$$

$$=32500$$

$$\sum y_iV_i=30(150)+10(350)+70(100)+50(400)$$

$$=35000$$

$$\sum V_i=150+350+100+400=1000$$

$$x_c =\dfrac{32500}{1000}=32.5$$

$$y_c =\dfrac{35000}{1000}=35$$

The weighted center of gravity is located at  $x_c=32.5, y_c=35.$

$$\sum x_i=30+20+10+50=110$$

$$\sum y_i=30+10+70+50=160$$

The simple center of gravity is located at  $x_s=\dfrac{110}{4}=27.5, y_s=\dfrac{160}{4}=40.$