You are given x is directly proportional to y^3, and y is inversely proportional to sqrt{z}. If the value of x is 3 when z is 12, what is the value of x when z is equal to 75?
+9466 x is directly proportional to y3 so \(x=ky^3\) where k is a constant.
y is inversely proportional to √z so \(y=\dfrac{s}{\sqrt{z}}\) where s is a constant.
\(x=ky^3\\~\\ x=k\Big(\frac{s}{\sqrt{z}}\Big)^3 \\~\\ x=\frac{ks^3}{z\sqrt{z}}\)
When z = 12 , x = 3 So...
\(3=\frac{ks^3}{12\sqrt{12}}\\~\\ ks^3= 3\cdot12\sqrt{12}\\~\\ks^3=72\sqrt3\)
When z = 75 ,
\(x=\frac{ks^3}{z\sqrt{z}}\\~\\ x=\frac{72\sqrt3}{75\sqrt{75}}\\~\\ x=\frac{72\sqrt3}{375\sqrt{3}}\\~\\ x=\frac{24}{125}\)
+36916 X = k y^3 y = k/sqrtz combine to
x = k /sqrt z ^3
or
xsqrt z^3 = k always
3(sqrt12)^3 = xsqrt( 75) ^3
x = 3 (sqrt12)^3 /(sqrt75)^3
x = 3 *8 /125 = 24/125
+9466 x is directly proportional to y3 so \(x=ky^3\) where k is a constant.
y is inversely proportional to √z so \(y=\dfrac{s}{\sqrt{z}}\) where s is a constant.
\(x=ky^3\\~\\ x=k\Big(\frac{s}{\sqrt{z}}\Big)^3 \\~\\ x=\frac{ks^3}{z\sqrt{z}}\)
When z = 12 , x = 3 So...
\(3=\frac{ks^3}{12\sqrt{12}}\\~\\ ks^3= 3\cdot12\sqrt{12}\\~\\ks^3=72\sqrt3\)
When z = 75 ,
\(x=\frac{ks^3}{z\sqrt{z}}\\~\\ x=\frac{72\sqrt3}{75\sqrt{75}}\\~\\ x=\frac{72\sqrt3}{375\sqrt{3}}\\~\\ x=\frac{24}{125}\)
