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geometry senior
Problem
The diameter of a circle is divided into equal parts. On each part a semicircle is constructed. As becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length:
(A)
equal to the semi-circumference of the original circle
(B)
equal to the diameter of the original circle
(C)
greater than the diameter, but less than the semi-circumference of the original circle
(D)
that is infinite
Solution
Note that the half the circumference of a circle with diameter is . Let's call the diameter of the circle D. Dividing the circle's diameter into n parts means that each semicircle has diameter , and thus each semicircle measures . The total sum of those is , and since that is the exact expression for the semi-circumference of the original circle, the answer is .
Final answer
A