Whenever the owners of 50% or more of the area in any district shall present to the Board of Supervisors a petition, duly signed and acknowledged, requesting an amendment, supplement, change, modification or repeal of any of the regulations or restrictions prescribed by this chapter for their district or a change or modification of the Zoning Map with reference to such district, it shall be the duty of the Board of Supervisors to hold a public hearing thereon and cause notice thereof to be given in the manner prescribed in § 230-188 above.