John wants to measure the width of a river. He marks off two right triangles, as shown in the figure. The base of the larger triangle has a length of $55 \mathrm{~m}$, and the base of the smaller triangle has a length of $31 \mathrm{~m}$. The height of the smaller triangle is $22.7 \mathrm{~m}$. How wide is the river? Round your answer to the nearest meter. (The figure is not drawn to scale.)

Step 1 ：Given that John marks off two right triangles, the base of the larger triangle has a length of \(55 \mathrm{m}\), and the base of the smaller triangle has a length of \(31 \mathrm{m}\). The height of the smaller triangle is \(22.7 \mathrm{m}\).

Step 2 ：Since the two triangles are similar, the ratio of the corresponding sides of similar triangles is equal. Therefore, we can set up a proportion to solve for the unknown side. The width of the river corresponds to the height of the larger triangle.

Step 3 ：We can set up the proportion as follows: \[\frac{base_{small}}{height_{small}} = \frac{base_{large}}{height_{large}}\]

Step 4 ：Substituting the given values into the equation, we get: \[\frac{31}{22.7} = \frac{55}{height_{large}}\]

Step 5 ：Solving for the height of the larger triangle, which is the width of the river, we get: \[height_{large} = \frac{55 \times 22.7}{31} \approx 40.274193548387096\]

Step 6 ：Rounding to the nearest meter, we get \(\boxed{40}\) meters.