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The surface area of a cube is $6\\left(\\dfrac{a}{4}\\right)^{2}$, where $a$ is a positive constant. Which of the following gives the perimeter of one face of the cube?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-134"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

The surface area of a cube is $6\\left(\\dfrac{a}{4}\\right)^{2}$, where $a$ is a positive constant. Which of the following gives the perimeter of one face of the cube?

","comment":{"@type":"Comment","text":"

Choice B is correct. A cube has $6$ faces of equal area, so if the total surface area of a cube is $6\\left(\\dfrac{a}{4}\\right)^{2}$, then the area of one face is $\\left(\\dfrac{a}{4}\\right)^{2}$. Likewise, the area of one face of a cube is the square of one of its edges; therefore, if the area of one face is $\\left(\\dfrac{a}{4}\\right)^{2}$, then the length of one edge of the cube is $\\dfrac{a}{4}$. Since the perimeter of one face of a cube is four times the length of one edge, the perimeter is $4\\left(\\dfrac{a}{4}\\right)=a$.

Choice A is incorrect because if the perimeter of one face of the cube is $\\dfrac{a}{4}$,

then the total surface area of the cube is $6\\left(\\frac{a}{\\frac{4}{4}}\\right)^{2} =6\\left(\\frac{a}{16}\\right)^{2}$, which is not $6\\left(\\dfrac{a}{4}\\right)^{2}$. Choice C is incorrect because if the perimeter of one face of the cube is $4a$, then the total surface area of the cube is $6\\left(\\dfrac{4a}{4}\\right)^{2}=6a^{2}$, which is not $6\\left(\\dfrac{a}{4}\\right)^{2}$.

Choice D is incorrect because if the perimeter of one face of the cube is $6a$, then the total surface area of the cube is $6\\left(\\dfrac{6a}{4}\\right)^{2}=6\\left(\\dfrac{3a}{2}\\right)^{2}$, which is not $6\\left(\\dfrac{a}{4}\\right)^{2}$

"},"encodingFormat":"text/markdown","text":"

The surface area of a cube is $6\\left(\\dfrac{a}{4}\\right)^{2}$, where $a$ is a positive constant. Which of the following gives the perimeter of one face of the cube?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$\\dfrac{a}{4}$

"},{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$4a$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$6a$

"}],"acceptedAnswer":{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$a$

","comment":{"@type":"Comment","text":"

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-134"}

The surface area of a cube is $6\left(\dfrac{a}{4}\right)^{2}$, where $a$ is a positive constant. Which of the following gives the perimeter of one face of the cube?