1. The problem states that $P$ is proportional to $\frac{1}{2}H$, and we are given $P=2$ when $G=4$ and $H=3$. We need to find $H$ when $P=3$ and $G=8$. 2. Since $P \propto \frac{1}{2}H$, we can write this as an equation with a constant of proportionality $k$: $$P = k \times \frac{1}{2}H$$ 3. Using the given values $P=2$ and $H=3$, substitute to find $k$: $$2 = k \times \frac{1}{2} \times 3$$ $$2 = \frac{3k}{2}$$ 4. Solve for $k$: $$k = \frac{2 \times 2}{3} = \frac{4}{3}$$ 5. Now, the formula for $P$ is: $$P = \frac{4}{3} \times \frac{1}{2}H = \frac{2}{3}H$$ 6. We want to find $H$ when $P=3$: $$3 = \frac{2}{3}H$$ 7. Solve for $H$: $$H = 3 \times \frac{3}{2} = \frac{9}{2} = 4.5$$ 8. The value of $H$ is $4.5$, which corresponds to option C) 4 (closest to 4.5). However, since 4.5 is not exactly 4, none of the options exactly match. If the options are approximate, the closest is 4. Final answer: $H = \frac{9}{2} = 4.5$