1. **Problem Statement:** We start with 100 liters of water and 5 kilograms of sugar in a tank. Water is added at 10 liters per minute and sugar at 2 kilograms per minute. We want to find the concentration $C(t)$ of sugar water in kilograms per liter at time $t \geq 0$ minutes. 2. **Formula for concentration:** Concentration is defined as the amount of sugar divided by the total volume of the solution: $$C(t) = \frac{\text{amount of sugar at time } t}{\text{total volume at time } t}$$ 3. **Calculate amount of sugar at time $t$:** Initial sugar = 5 kg Sugar added per minute = 2 kg/min So, sugar at time $t$ is: $$5 + 2t$$ 4. **Calculate total volume at time $t$:** Initial volume = 100 liters Water added per minute = 10 liters/min So, volume at time $t$ is: $$100 + 10t$$ 5. **Write the expression for concentration:** $$C(t) = \frac{5 + 2t}{100 + 10t}$$ 6. **Interpretation:** This expression shows how the concentration changes over time as more water and sugar are added. 7. **Answer choice:** Comparing with the options, the correct expression is option D: $$\boxed{\frac{5 + 2t}{100 + 10t}}$$