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Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = -3$ when $\beta = -6$, find $\alpha$ when $\beta = 18$. Express your answer as a fraction.

 Apr 18, 2022
 #1
avatar+578 
+2

\(\frac{-3}{6}=\frac{a}{18}\quad \)

 

\(\frac{a}{18}=-\frac{1}{2}\)

 

\(:\quad a=-9\)

 

 

-Vinculum

 

smileysmileysmiley

 Apr 18, 2022
 #2
avatar+9459 
+2

You have assumed \(\alpha\) is directly proportional to \(\beta\) instead of inversely proportional to \(\beta\).

MaxWong  Apr 19, 2022
 #3
avatar+9459 
+1

Since \(\alpha\) is inversely proportional to \(\beta\)\(\alpha = \dfrac k\beta\) for some nonzero real number \(k\), which we call the proportionality constant.

We find k by substituting a pair of values of \(\alpha\) and \(\beta\) into the equation.

 

Substituting \(\alpha = -3\) and \(\beta = -6\)\(-3 = \dfrac k{-6}\). Solving gives \(k = 18\).

Now, substitute \(\beta = 18\) and \(k = 18\) to get the corresponding value of \(\alpha\) when \(\beta = 18\).

\(\alpha = \dfrac{18}{18} = \boxed{1}\).

 Apr 19, 2022

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