23. Four concentric circles have radii 10, 20, 30, and 40. What is the ratio of the sum of the areas of the shaded annuli (rings between circles) to the area of the largest circle?
Answer: B
The ratio of the sum of the areas of the shaded annuli to the area of the largest circle is 3:4.
The shaded annuli are formed between the concentric circles with radii 10, 20, 30, and 40. The area of the largest circle is calculated as π(40^2), and the total area of the annuli corresponds to the area of the largest circle minus the areas of the smaller circles, leading to a ratio of 3:4.
A) 1:4
This option is incorrect because the ratio of the sum of the shaded areas to the area of the largest circle does not simplify to 1:4. The total shaded area is significantly more than one-fourth of the area of the largest circle, as it accounts for multiple annular sections.
B) 3:4
This is the correct option. The area of the largest circle is π(40^2) = 1600π, and the areas of the annuli (the differences between successive circles' areas) total to 1200π, which gives a ratio of 1200π:1600π, simplifying to 3:4.
C) 1:2
This option is incorrect because the ratio of the shaded areas to the area of the largest circle is greater than 1:2. The calculation reveals that the shaded area is not half of the largest circle's area, indicating a misunderstanding of the total shaded annular area.
D) 2:3
This option is incorrect as well. The ratio of the shaded areas to the area of the largest circle exceeds 2:3 based on the calculated values. The total shaded area indicates a more substantial relationship to the largest circle's area than is represented by this ratio.
Conclusion
The correct ratio of 3:4 indicates that the areas of the shaded annuli constitute a significant portion of the area of the largest circle. Other options fail as they underestimate the relationship between the shaded areas and the total area of the largest circle, demonstrating a clear discrepancy in their calculations.