(a) A county from which a person is committed to the Department of Corrections and Rehabilitation, Division of Juvenile Facilities, shall pay to the state an annual rate of twenty-four thousand dollars ($24,000) while the person remains in an institution under the direct supervision of the division, or in an institution, boarding home, foster home, or other private or public institution in which the person is placed by the division, and cared for and supported at the expense of the division, as provided in this subdivision. This subdivision applies to a person who is committed to the division by a juvenile court on or after July 1, 2012. The Department of Corrections and Rehabilitation, Division of Juvenile Facilities, shall present to the county, not more frequently than monthly, a claim for the amount due to the state under this subdivision, which the county shall process and pay pursuant to Chapter 4 (commencing with Section 29700) of Division 3 of Title 3 of the Government Code.