1. **State the problem:** We are given that $T$ is inversely proportional to $w$, and $w$ is directly proportional to the cube root of $d$. We have values for $w$, $T$, and $d$ at certain points and need to find $d$ when $T=36$. 2. **Write the proportionality equations:** Since $T$ is inversely proportional to $w$, we write: $$T = \frac{k}{w}$$ for some constant $k$. Since $w$ is directly proportional to the cube root of $d$, we write: $$w = m \sqrt[3]{d}$$ for some constant $m$. 3. **Use given values to find constants:** When $w=9$, $T=10$: $$10 = \frac{k}{9} \implies k = 90$$ When $w=1$, $d=64$: $$1 = m \sqrt[3]{64} = m \times 4 \implies m = \frac{1}{4}$$ 4. **Express $T$ in terms of $d$:** Substitute $w = \frac{1}{4} \sqrt[3]{d}$ into $T = \frac{90}{w}$: $$T = \frac{90}{\frac{1}{4} \sqrt[3]{d}} = 90 \times \frac{4}{\sqrt[3]{d}} = \frac{360}{\sqrt[3]{d}}$$ 5. **Find $d$ when $T=36$:** $$36 = \frac{360}{\sqrt[3]{d}} \implies \sqrt[3]{d} = \frac{360}{36} = 10$$ Cube both sides: $$d = 10^3 = 1000$$ **Final answer:** $$d = 1000$$