In one of the warehouses of a construction firm, cement is being released from an elevated storage bin at a rate of 29 ft³/min. A wide and flat metallic sheet below the bin catches the cement which forms a conical pile. The base diameter and the height of the pile is maintained at a 1:1 ratio. Find the rate at which the area of the base of the cone is increasing when the diameter is 14 feet. Provide both the working equation and the differentiated equation of this problem; include this in the summary of answers.

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