1. **Problem Statement:** The total cost $C$ to build a wooden fence varies jointly with the length $L$ of the fence and the square root of the wood quality index $Q$. Given a 10-meter fence with wood quality index 9 costs 1200, find the cost for a 15-meter fence with wood quality index 4. 2. **Write the joint variation equation:** Since $C$ varies jointly with $L$ and $\sqrt{Q}$, the formula is: $$C = k L \sqrt{Q}$$ where $k$ is the constant of proportionality. 3. **Find the constant $k$:** Substitute the known values $C=1200$, $L=10$, and $Q=9$: $$1200 = k \times 10 \times \sqrt{9}$$ $$1200 = k \times 10 \times 3$$ $$1200 = 30k$$ Solve for $k$: $$k = \frac{1200}{30} = 40$$ 4. **Write the equation with $k$:** $$C = 40 L \sqrt{Q}$$ 5. **Find the cost for $L=15$ and $Q=4$:** $$C = 40 \times 15 \times \sqrt{4}$$ $$C = 40 \times 15 \times 2$$ $$C = 40 \times 30 = 1200$$ **Final answer:** The total cost to build the 15-meter fence with wood quality index 4 is $1200$. This shows how joint variation works: the cost depends on both length and the square root of quality, and we find the constant from given data before calculating the unknown cost.