if $y$ varies directly as $x$, and $y$ is 400 when $x$ is $r$ and $y$ is $r$ when $x$ is 4, what is the numeric constant of variation in the relation?

10
40
100
198

Question

if $y$ varies directly as $x$, and $y$ is 400 when $x$ is $r$ and $y$ is $r$ when $x$ is 4, what is the numeric constant of variation in the relation?

10
40
100
198

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Direct Variation" ], "new_concepts": [], "current_concepts": [ "Direct Variation", "Solving Quadratic Equations" ] } ## Set up the direct variation equations $$ \begin{aligned} y &= kx \ 400 &= k \cdot r \ r &= k \cdot 4 \end{aligned} $$ ## Solve for the constant of variation $$ \begin{aligned} r &= 4k \ 400 &= k(4k) \ 400 &= 4k^2 \ k^2 &= 100 \ k &= 10 \quad (\text{since } k > 0 \text{ for the given options}) \end{aligned} $$

(A) 10

(B) 40

(C) 100

(D) 198

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Direct Variation" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the direct variation equations

Solve for the constant of variation

Answer: