II. (25 points) A steamboat tows several small boats, each fully loaded with equal weight of goods, to transport goods between two ports. It is known that when towing 4 small boats each time, it can make 16 round trips in a day; when towing 7 small boats each time, it can make 10 round trips in a day. The number of round trips per day is a linear function of the number of small boats towed (the number of small boats towed each time remains constant throughout the day). How many times per day should it make round trips, and how many small boats should it tow each time, to maximize the total amount of goods transported?
Let's assume that $x$ boats are towed each time every day, and the round trips are made $y$ times a day, with each boat having a carrying capacity of $M$. If $y=k x+b$, then
$$
\left\{\begin{array}{l}
4 k+b=16 \\
7 k+b=10 .
\end{array}\right.
$$

Solving these equations, we get $k=-2, b=24$, which means $y=-2 x+24$.
Thus, the total cargo weight transported each day is
$$
G=M x y=-2 M(x-6)^{2}+72 M \text {. }
$$

When $x=6, y=12$, $G$ reaches its maximum value, which is $72 M$.
That is, making 12 round trips each day, and towing 6 boats each time, results in the maximum cargo weight transported.