problems about variation

Q: Suppose y varies directly as x^2 and z varies directly as x^3, then........

y=k’x^2, k’不等於0
y^3=k’^3 x^6

z=k”x^3, k”不等於0
z^2=k”^2 x^6

y^3/z^2=(k’^3 x^6)/(k”^2 x^6)
y^3/z^2=k’^3/k”^2= k,where k=k’^3/k”^2 is constant

so,y^3 varies directly as z^2.

Q: The price of a cylindrical rod of radius r and length h varies directly as the volume of the rod. When r=3cm and h=8cm, the price is $20. Find the price of the rod when r=6cm and h=12cm.

p=k(pi)r^2*h, k不等於0

When r=3,h=8,p=20
20=72*(pi)*k
k=5/18(pi)

so,p=(5/18)*r^2*h

When r=6,h=12
p=(5/18)*(6)^2*(12)
p=120

sothe price of the rod is $ 120 when r=6cm and h=12cm

y = kx^2, that is y^3 = k^3x^6
z = hx^3, that is z^2 = h^2x^6
so y^3/z^2 = k^3/h^2 which is also a constant, therefore, y^3 varies directly as z^2.