P varies directly as square of Q. If P = 3 and Q = 2, find Q when P = 27.

If $P$ varies directly as the square of $Q$, the relationship can be written as:

$$P = kQ^2$$

where $k$ is the constant of variation.

We are given $P=3$ when $Q=2$. Substitute these values into the equation:

$$3 = k(2^2)$$ $$3 = k(4)$$

To find $k$, divide both sides by 4:

$$k = \frac{3}{4}$$

Now use the constant $k = \frac{3}{4}$ and the new value $P=27$ in the variation equation:

$$27 = \frac{3}{4}Q^2$$

To solve for $Q^2$, multiply both sides by $\frac{4}{3}$:

$$Q^2 = 27 \times \frac{4}{3}$$ $$Q^2 = 9 \times 4$$ $$Q^2 = 36$$

Take the square root of both sides to find $Q$:

$$Q = \pm\sqrt{36}$$ $$Q = \pm 6$$

The possible values for $Q$ are 6 and $-6$.