36. In the figure shown, all line segments meet at right angles, and the area of each rectangular region is one-half the area of the next larger region. What fractional part of square region ABCD is shaded?
Answer: C
The fractional part of square region ABCD that is shaded is 17/64.
The shaded area of square region ABCD is represented as 17/64 of the total area. This indicates that the shaded regions occupy a significant but specific proportion of the entire square.
A) 5/16
This option suggests that 5/16 of the total area is shaded. However, this is inconsistent with the given relationship of areas between the rectangular regions, which indicates that the shaded area is smaller than this fraction, making it incorrect.
B) 9/32
While 9/32 is a fraction that could potentially be considered, it does not align with the relationship of the areas described in the problem. The areas decrease in a specific ratio, and 9/32 does not fit within the pattern established, rendering it an incorrect choice.
C) 17/64
This option accurately reflects the relationship between the shaded area and the total area of square region ABCD. Given that each rectangular region's area is half of the next larger region, 17/64 is consistent with this progression and correctly represents the shaded portion.
D) 33/128
This choice suggests that 33/128 of the area is shaded, which is too small compared to the actual fractional part that is shaded. The ratio of areas defined in the problem indicates that the shaded area must be larger than this fraction, making it an incorrect option.
Conclusion
The correct answer, 17/64, reflects the precise calculation of the shaded area in relation to the whole square. The other options do not correspond to the established area relationships among the rectangular regions, which confirms that they are incorrect. Thus, 17/64 is definitively the correct fractional part of square region ABCD that is shaded.