1952 AMC 12 Problem 43

Question

The diameter of a circle is divided into

n

equal parts. On each part a semicircle is constructed. As

n

becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length:

Equal to the semi-circumference of the original circle

Equal to the diameter of the original circle

Greater than the diameter but less than the semi-circumference of the original circle

That is infinite

Greater than the semi-circumference but finite

Let the original diameter be

d

, so each small semicircle has diameter

\frac{d}{n}

Multiply the arc length of one small semicircle by the number

n

of parts

Each small semicircle has diameter

\frac{d}{n},

so its arc length is

\frac{\pi d}{2n}.

The sum of all

n

arc lengths is

n\cdot\frac{\pi d}{2n}=\frac{\pi d}{2},

exactly the semi-circumference of the original circle. This equality holds for every positive

n,

not merely in the limit.

Thus, the correct answer is A.

Problem 43 in other years