Answer
1620 $ft^{2}$
Work Step by Step
Given in a regular dodecagon the approximate ratio of the length of an apothem to the length of a side is 15:8
For a regular dodecagon with a side of length 12 ft
The area of a regular polygon whose apothem has length a and whose perimeter P is given by A = $\frac{1}{2}$ ap
Lets take the ratio of apothem and side ratio 15:8. By proportionality constant x
The apothem a = 15x
side s= 8x
But given side of length = 12 ft
s = 8x = 12ft
x = $\frac{12}{8}$ = 1.5 ft
apothem of dodecagon a = 15x = 15* 1.5ft= 22.5 ft
The perimeter of dodecagon P = 12s = 12 * 12 = 144 ft
The area of octagon A = $\frac{1}{2}$ ap
= $\frac{1}{2}$ *22.5 * 144 = 1620 $ft^{2}$