1. **State the problem:** We have a tank with 100 gallons of water at noon. Pipes A and B add water to the tank, while pipe C removes water. We want to find the function $T(x)$ that represents the amount of water in the tank $x$ minutes after the pipes start operating. 2. **Identify the given functions:** - Pipe A flow in: $a(x) = 25x$ - Pipe B flow in: $b(x) = 10x$ - Pipe C flow out: $c(x) = 30x$ 3. **Understand the problem:** - The tank starts with 100 gallons. - Water added by pipes A and B increases the tank volume. - Water removed by pipe C decreases the tank volume. 4. **Write the general formula:** $$T(x) = \text{initial amount} + \text{total inflow} - \text{total outflow}$$ 5. **Substitute the given functions:** $$T(x) = 100 + a(x) + b(x) - c(x)$$ 6. **Simplify the expression:** $$T(x) = 100 + 25x + 10x - 30x$$ $$T(x) = 100 + (25x + 10x - 30x)$$ $$T(x) = 100 + 5x$$ 7. **Interpretation:** The amount of water in the tank increases by 5 gallons per minute after the pipes start operating. **Final answer:** $$T(x) = 100 + 25x + 10x - 30x$$ This corresponds to option C.