x varies directly as the square of y and inversely as the cube root of z and x = 2, when y = 4, z= 8. What is the value of y when x = 3, and z = 27?

Given:

Direct variation relationship = x ∝ y².

Inverse variation relationship = x ∝ 1/z^(1/3).

Concept used:

In variation problems, establishing the constant of proportionality allows transformation into equality.

Calculations:

x = k (y²/z^(1/3)) where k is the constant of proportionality.

Substituting x = 2, y = 4, and z = 8

⇒  2 = k((4²) ÷ (8^(1/3)) 

⇒ k = 0.25

Now substituting x = 3, z = 27, and k = 0.25 

⇒ 3 = 0.25(y²/(27^(1/3))).

⇒ 3 = 0.25(y²/3)

⇒ y² = 900/25

⇒  y² = ±36.

⇒ y = ±6.

∴ y can be either 6 or -6.