Work out the value of R when P = 0.5.

Steps to Solve

1. Determine the relationship between P and Q Since $P$ is directly proportional to $Q$, we can express this as: $$ P = k_1 Q $$ where $k_1$ is a constant.

Given that when $P = 8$ and $Q = 2$, we substitute these values to find $k_1$: $$ 8 = k_1 \cdot 2 $$ Thus, $$ k_1 = \frac{8}{2} = 4 $$

2. Establish the relationship between R and Q Since $R$ is inversely proportional to $Q^2$, we can write: $$ R = \frac{k_2}{Q^2} $$ where $k_2$ is another constant.

Using the values when $R = 10$ and $Q = 3$, we find $k_2$: $$ 10 = \frac{k_2}{3^2} $$ Thus, $$ k_2 = 10 \cdot 9 = 90 $$

3. Finding Q when P = 0.5 Now we need to find the value of $Q$ when $P = 0.5$. Using our relationship from step 1: $$ 0.5 = 4 Q $$ Solving for $Q$ gives us: $$ Q = \frac{0.5}{4} = 0.125 $$

4. Calculate R using the new Q value Now we substitute $Q = 0.125$ into the equation for $R$ from step 2: $$ R = \frac{90}{(0.125)^2} $$ Calculating $(0.125)^2$: $$ (0.125)^2 = 0.015625 $$ Hence, $$ R = \frac{90}{0.015625} = 5760 $$