Question

suppose y varies directly with the square root of x and inversely with z. when x=4 and z=3, then y=10. find y if x=49 and z=3

Answer 1

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

$\bf \begin{array}{llll} \textit{y varies directly with}\\ \textit{the square root of x and inversely with z} \end{array}\qquad y=\cfrac{k\sqrt{x}}{z} \\\\\\ \textit{we also know that } \begin{cases} x=4\\ z=3\\ y=10 \end{cases}\implies 10=\cfrac{k\sqrt{4}}{3}\implies 30=k2 \\\\\\ \cfrac{30}{2}=k\implies 15=k\qquad \qquad \boxed{y=\cfrac{15\sqrt{x}}{z}} \\\\\\ \textit{when x = 49 and z = 3, what is \underline{y}?}\qquad \qquad y=\cfrac{15\sqrt{49}}{3}$

$\bf y=\cfrac{15\cdot 7}{3}\implies y=\cfrac{15}{3}\cdot 7\implies y=5\cdot 7\implies y=35$