Upon receiving a certified copy of the resolution, if the board of supervisors finds by four-fifths vote of all its members that the enlargement or improvement of the public park, beach, golf course, or recreation ground is of general county interest or that the cost of maintenance is increased by reason of use by residents of the county outside of the city, it may determine to extend aid to the city for the purpose in an amount to be fixed by it by resolution, but not exceeding the amount requested by the legislative body of the city.