10. (12 points) There are two docks, $A$ and $B$, on a river, with $A$ upstream and $B$ downstream. There are two boats, Boat A and Boat B, where Boat A's speed in still water is twice that of Boat B. Boat A and Boat B start from docks $A$ and $B$ respectively at the same time, heading towards each other. When Boat A starts, it leaves a box that can float on the water surface. After 20 minutes, the two boats meet, and at this moment, Boat A leaves another box on the water surface. After some time, Boat A realizes it has lost its cargo and turns back to search for it. When Boat A finds the second box, Boat B happens to encounter the first box left by Boat A. How many minutes did it take for Boat A to realize it had lost its cargo from the time it started? (Neglect the time for turning around.)


【Solution】Solution: Let the time after the two boats meet and before Boat A discovers its cargo is missing be $x$ minutes.
During this time, the distance between Boat A and the second cargo is: $x \times (V_{\text{A}} + V_{\text{water}}) - x \times V_{\text{water}} = V_{\text{A}} x$. After this, Boat A turns around to look for the second cargo, so the distance Boat A and the second cargo need to meet is also $V_{\text{A}} x$. According to the formula for meeting time $=$ total distance $\div$ sum of speeds, the time for Boat A and the second cargo to meet is $V_{\text{A}} x \div (V_{\text{A}} - V_{\text{water}} + V_{\text{water}}) = x$,
which means that from the time Boat A discovers the second cargo is missing to the time it finds the second cargo, it takes a total of $x + x = 2x$ minutes. During this $2x$ minutes, Boat B and the first cargo meet. The distance Boat B and the first cargo meet is the distance Boat A travels more than the first cargo in 20 minutes, which is $20 \times (V_{\text{A}} + V_{\text{water}}) - 20 \times V_{\text{water}} = 20 V_{\text{A}}$, so $2x \times (V_{\text{B}} - V_{\text{water}} + V_{\text{water}}) = 20$. Since $V_{\text{A}} = 2 V_{\text{B}}$, then $x = 20$, $20 + 20 = 40$ (minutes).
Answer: Boat A discovers its cargo is missing 40 minutes after departure.