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:

• 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-circle
• D. that is infinite
• E. greater than the semi-circumference but finite

Answer 1

The correct option is A equal to the semi-circumference of the original circle

Length of the diameter = 2r

Length of radius of each smaller semicircle = $\frac{r}{n}$

Length of each arc = $\frac{\pi r}{n}$

Sum of the lengths of arcs = $n\left(\frac{\pi r}{n}\right)=\pi r$

Even as n becomes large, sum of the lengths of the arcs = $\pi r$ = semi circumference of the original circle