Problem:
An even number of circles are nested, starting with a radius of 1 and increasing by 1 each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius 2 but outside the circle of radius 1. An example showing 8 circles is displayed below. What is the least number of circles needed to make the total shaded area at least 2023Ï€?
Answer Choices:
A. 46
B. 48
C. 56
D. 60
E. 64
Solution:
Suppose there are 2n circles. The shaded area is then
To find the least value of n such that A≥2023π, note that n≈22023​​≈32. Because 31(2⋅31+1)= 1953 and 32(2⋅32+1)=2080, the required value is n=32, and there are 2⋅32=(E)64​ circles.
