1. **State the problem:** A rectangular tank is 10 m long and 5 m wide. It is initially 45% filled with juice. After adding 75 m^3 of water, the tank becomes 3/5 (or 60%) full. We need to find the height $h$ of the tank. 2. **Define variables and known values:** - Length $L = 10$ m - Width $W = 5$ m - Height $h$ (unknown) - Initial volume of juice $= 45\%$ of total volume $= 0.45 \times L \times W \times h$ - Volume after adding water $= 60\%$ of total volume $= 0.60 \times L \times W \times h$ - Volume of water added $= 75$ m$^3$ 3. **Set up the equation:** The volume after adding water equals the initial volume plus 75 m$^3$: $$ 0.60 \times L \times W \times h = 0.45 \times L \times W \times h + 75 $$ 4. **Simplify the equation:** $$ 0.60 \times 10 \times 5 \times h = 0.45 \times 10 \times 5 \times h + 75 $$ $$ 0.60 \times 50 \times h = 0.45 \times 50 \times h + 75 $$ $$ 30h = 22.5h + 75 $$ 5. **Solve for $h$:** $$ 30h - 22.5h = 75 $$ $$ 7.5h = 75 $$ $$ h = \frac{75}{7.5} = 10 $$ 6. **Interpretation:** The height of the tank is 10 meters. **Final answer:** $$h = 10 \text{ meters}$$