1. **State the problem:** We need to find the compound interest rate $r$ such that an initial principal $P=5000$ grows to a final amount $A=9000$ in $t=4$ years. 2. **Formula used:** The compound interest formula is $$A = P \left(1 + \frac{r}{100}\right)^t$$ where $r$ is the annual interest rate in percent. 3. **Substitute known values:** $$9000 = 5000 \left(1 + \frac{r}{100}\right)^4$$ 4. **Isolate the compound factor:** Divide both sides by 5000: $$\frac{9000}{5000} = \left(1 + \frac{r}{100}\right)^4$$ 5. **Simplify the fraction:** $$\cancel{\frac{9000}{5000}} = \frac{9}{5} = 1.8 = \left(1 + \frac{r}{100}\right)^4$$ 6. **Take the fourth root of both sides:** $$\sqrt[4]{1.8} = 1 + \frac{r}{100}$$ 7. **Calculate the fourth root:** $$\sqrt[4]{1.8} \approx 1.1574$$ 8. **Solve for $r$:** $$1.1574 = 1 + \frac{r}{100} \implies \frac{r}{100} = 0.1574 \implies r = 15.74$$ **Final answer:** The required compound interest rate is approximately **15.74%** per annum.