1. **State the problem:** We are given that $b$ is inversely proportional to the cube of $a$, and we need to find the constant of proportionality $k$ when $b=60$ and $a=5$. 2. **Write the formula:** Since $b$ is inversely proportional to the cube of $a$, we have $$b = \frac{k}{a^3}$$ where $k$ is the constant of proportionality. 3. **Substitute the known values:** Plug in $b=60$ and $a=5$ into the formula: $$60 = \frac{k}{5^3}$$ 4. **Simplify the denominator:** $$60 = \frac{k}{125}$$ 5. **Solve for $k$ by multiplying both sides by 125:** $$60 \times 125 = \cancel{125} \times \frac{k}{\cancel{125}}$$ $$7500 = k$$ 6. **Final answer:** The constant of proportionality is $$\boxed{7500}$$