Let $D$ Be a Region Bounded by a Simple Closed Path $C$

question 103

Multiple Choice

Let $D$ be a region bounded by a simple closed path $C$ in the $x y$ . Then the coordinates of the centroid $( \bar { x } , \bar { y } )$ of $D$ are $\bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ where $A$ is the area of $D$ .
Find the centroid of the triangle with vertices $( 0,0 ) , ( 5,0 )$ and $( 0,10 )$ .

• A) \(\left( \frac { 1 } { 8 } , \frac { 1 } { 8 } \right)\)
• B) \(\left( \frac { 10 } { 3 } , \frac { 10 } { 3 } \right)\)
• C) \(\left( \frac { 1 } { 10 } , \frac { 1 } { 5 } \right)\)
• D) \(\left( \frac { 5 } { 3 } , - \frac { 10 } { 3 } \right)\)
• E) \(\left( \frac { 1 } { 8 } , \frac { 1 } { 64 } \right)\)

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Definitions:

Working Memory

The part of short-term memory that is concerned with immediate, conscious perceptual and linguistic processing.

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In mathematics, the relationship or expression that associates each element of a set with exactly one element of another set. In general use, it refers to the natural purpose or activities specific to a person, thing, or institution.

Nativist Theory

A perspective suggesting that certain skills or abilities are "native" or hard-wired into the brain at birth.

Language Acquisition

The process by which humans gain the ability to perceive, produce, and understand language, often focusing on the developmental stages in early childhood.