Question

The quantity y varies directly with the square of x and inversely with z. When x is 9 and z is 27, y is 6. What is the constant of variation?


2

18

54

Answer 1

$\bf \qquad \qquad \textit{double proportional variation} \\\\ \begin{array}{llll} \textit{\underline{y} varies directly with \underline{x}}\\ \textit{and inversely with \underline{z}} \end{array}\implies y=\cfrac{kx}{z}\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------$

$\bf \stackrel{\textit{\underline{y} varies directly with the square of \underline{x} and inversely with \underline{z}}}{y=\cfrac{kx^2}{z}} \\\\\\ \textit{we also know that } \begin{cases} x=9\\ z=27\\ y=6 \end{cases}\implies 6=\cfrac{k9^2}{27} \\\\\\ \cfrac{27\cdot 6}{9^2}=k\implies 2=k$