The answer to the third question on page 69 of the Nine Heroes Mathematics Book published by Beijing Normal University is urgent.

Let the distance from point B to the rectangle be x, and the hypotenuse of the triangle be 50. The height is 120/5.

The width of the rectangle is120/5-x.

The length of the rectangle is 50X*5/ 120.

S rectangle = 50x * 5/120 * (120/5-x)

=50X-250x^2/ 120

A is greater than 0, so the function has the maximum value, which is (4ac-b 2)/4a.

That is 50 2/(4 * 250/120) = 300.

This is just a method. You can also set the rectangular side to X and develop it yourself.