1. **Problem Statement:** We are given that $q$ is inversely proportional to $p^3$, and when $p=4$, $q=4$. We want to find the relationship between $q$ and $p$, and use it to find missing values in the table. 2. **Formula and Explanation:** If $q$ is inversely proportional to $p^3$, then: $$q = \frac{k}{p^3}$$ where $k$ is a constant. 3. **Find the constant $k$:** Given $p=4$ and $q=4$, substitute these values: $$4 = \frac{k}{4^3} = \frac{k}{64}$$ Multiply both sides by 64: $$k = 4 \times 64 = 256$$ 4. **Write the formula with $k$:** $$q = \frac{256}{p^3}$$ 5. **Use the formula to find missing values:** - When $p=5$: $$q = \frac{256}{5^3} = \frac{256}{125} = 2.048$$ - When $q=10$, find $p$: $$10 = \frac{256}{p^3} \implies p^3 = \frac{256}{10} = 25.6$$ Take cube root: $$p = \sqrt[3]{25.6} \approx 2.96$$ 6. **Summary:** - The formula relating $q$ and $p$ is $q = \frac{256}{p^3}$. - For $p=5$, $q \approx 2.048$. - For $q=10$, $p \approx 2.96$. This matches the table structure where some values are missing and can be calculated using this formula.