1. The problem states that a barrel already contains 10 gallons of water and is being filled with oil at a constant rate of 7.5 gallons per minute. 2. We want to write an expression for the amount of oil in the barrel after $t$ minutes, assuming the filling started at time $t=0$. 3. Since the barrel was initially filled with 10 gallons of water (not oil), the amount of oil at the start (when $t=0$) is $0$ gallons. 4. The rate of filling is 7.5 gallons per minute, so after $t$ minutes, the amount of oil added is $$7.5 \times t$$ gallons. 5. Therefore, the amount of oil in the barrel after $t$ minutes can be expressed as: $$\text{Oil}(t) = 7.5t$$ 6. Note that the total contents of the barrel at time $t$ is the initial 10 gallons of water plus the oil added: $$\text{Total volume}(t) = 10 + 7.5t$$ Final answer: - Oil volume after $t$ minutes is $$7.5t$$ gallons. - Total volume after $t$ minutes is $$10 + 7.5t$$ gallons.