P varies directly as the square of Q and inversely as R. If Q increases by 5% and R decreases by 10% find the percentage change in P.

P varies directly as the square of Q and inversely as R. This can be written as:

$$P = k \frac{Q^2}{R}$$

where $k$ is the constant of proportionality.

Q increases by 5%, so the new Q, $Q'$, is:

$$Q' = Q + 0.05Q = 1.05Q$$

R decreases by 10%, so the new R, $R'$, is:

$$R' = R - 0.10R = 0.9R$$

Substitute $Q'$ and $R'$ into the variation equation:

$$P' = k \frac{(Q')^2}{R'} = k \frac{(1.05Q)^2}{0.9R}$$ $$P' = k \frac{1.1025 Q^2}{0.9R}$$ $$P' = \frac{1.1025}{0.9} \left( k \frac{Q^2}{R} \right)$$

Since $P = k \frac{Q^2}{R}$, we have:

$$P' = 1.225 P$$

The percentage change is given by $\frac{P' - P}{P} \times 100\%$.

$$\text{Percentage change} = \frac{1.225P - P}{P} \times 100\%$$ $$\text{Percentage change} = \frac{0.225P}{P} \times 100\%$$ $$\text{Percentage change} = 0.225 \times 100\% = 22.5\%$$

Since the percentage change is positive, P increases by 22.5%.

Part (b):

Given $P = 2$, $R = 5$, and $Q = 4$.

$$P = k \frac{Q^2}{R}$$ $$2 = k \frac{4^2}{5}$$ $$2 = k \frac{16}{5}$$

Multiply both sides by 5:

$$10 = 16k$$

Divide by 16:

$$k = \frac{10}{16} = \frac{5}{8}$$

The relationship is now $P = \frac{5}{8} \frac{Q^2}{R}$.
Given $P = 4.5$ cm and $R = 5$ cm. We need to find the positive value of Q.

$$4.5 = \frac{5}{8} \frac{Q^2}{5}$$

Simplify the right side:

$$4.5 = \frac{Q^2}{8}$$

Multiply both sides by 8:

$$4.5 \times 8 = Q^2$$ $$36 = Q^2$$

Take the square root of both sides:

$$Q = \sqrt{36}$$

Since Q must be a positive value:

$$Q = 6$$

The final answers are:

a) The percentage change in P is $\boxed{22.5\% \text{ increase}}$.

b) The positive value of Q is $\boxed{6}$.