Math Problem Statement
y varies directly as the cube of x, and y = 49 when x = 7. Find the constant of variation and the equation for y.
Solution
The problemTo states solve that varies directly as the cube of , and the when . Let's solve for the let constant’s of analyze variation the ( statement k:
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y### varies Step directly-by as-step the solution cube: ofThe x general**, equation which for translates direct to variation the where equation (:
y\ $$)y varies = directly k asx the^ cube3 of (\ x]
\Here), is (:
k\[ )y is = the k constantx of^ variation3. We\ are]
given1: .- Substitute \(y y = = 4949\ )- and \(x x = 7
Step = 1: Substitute7 the given \ values) into into the equation:
[ 49 = k \cdot the equation \7[ ^493 = k(7^3$$
. Compute (]
Simplify 7^7^3:
343\[ 49\ =] So k,( the343 equation)$$ ### Step 2 becomes: $$:49 = k \cdot Solve for $$k$$ \[ k343 = \ frac{$$ 349}{.343} \ Solve for] Simplify \( the fraction k: \\): [ \k[ = \frac{1}{7} \] ### Final Answer: The constant of variation \(kk\ =) is: \ $$frack = \{frac49{}{1}{7} 343}$$ \] ### TheSimpl variationify the equation: \ fraction[ y: \ =[ k = \frac{1}{7} \] ### \ Finalfrac Answer: The{1}{7 constant of variation} is: $$x^k3 =\$$ fracWould you{ like1 further clarification or}{ have additional7 questions} \?] Here areThe some variation related equation topics is to explore: : \1[ y. What = \frac{ does1}{7}x^3 it mean when a\ variable] varies--- directly with anotherWould variable's you power like? 2. How a do detailed you simplify explanation of how fractions this in applies general? to other3 similar problems. What or have any are other types questions? of variation (###e.g., Related inverse Questions: ,1 joint.)? What happens to4 $$. y How \ does) the if cube \( function x$$ is doubled in affect this the growth variation rate equation of? a2 variable.? Can \(5. k Can \ you) graph be this negative equation in \(y = \ directfrac variation{ problems1? }{3.7 How}x^ do3\) you identify to the visualize type of the variation (direct, relationship inverse? ,** etcTip.):** from Always a word problem identify whether? the4 problem. describes What direct, would inverse, be or $$y \ combined variation to apply) the if appropriate \( formula x! = 2$$ in this specific equation? 5. What is the difference between direct and inverse variation? ### Tip: In direct variation, the constant $$k$$ always remains the same regardless of the values of $$x$$ and $$y$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Direct Variation
Exponents
Algebra
Formulas
y = k * x^3
k = y / x^3
Theorems
Direct variation theorem
Suitable Grade Level
Grades 8-10