Whenever any action is required or may be performed by any county officer or board for all purposes essential to the maintenance, operation, and administration of the district, such action shall be performed by the respective officer or board of the county of that part of the district in which the greatest amount of acreage of the district is located.

All costs incurred shall be borne proportionately by each county in that ratio which the amount of acreage of the district located in that part of each county forming a part of the district bears to the total amount of acreage located in the whole district.