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Difference between revisions of "1952 AHSME Problems/Problem 43"

(→‎Solution)
(The previous solution had the correct answer, but it didn't explain in any way the actual calculation to be done, in fact it had an error, and just bla bla bla with no justification, look for yourself to check it.)
 
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== Solution ==
 
== Solution ==
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Note that the circumference of a circle is <math>\pi*d</math>.  
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Note that the half the circumference of a circle with diameter <math>d</math> is <math>\frac{\pi*d}{2}</math>.  
  
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Let's call the diameter of the circle D. Dividing the circle's diameter into n parts means that each semicircle has diameter <math>\frac{D}{n}.</math> Since there are n circles, each with diameter <math>\frac{D}{n}</math>, the sum of the circumferences of the small circles is D. Since we are only drawing semicircles and not full circles, the requested sum is
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Let's call the diameter of the circle D. Dividing the circle's diameter into n parts means that each semicircle has diameter <math>\frac{D}{n}</math>, and thus each semicircle measures <math>\frac{D*pi}{n*2}</math>. The total sum of those is <math>n*\frac{D*pi}{n*2}=\frac{D*pi}{2}</math>, and since that is the exact expression for the semi-circumference of the original circle, the answer is <math>\boxed{A}</math>.
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<math>\boxed{A}</math>.
 
  
 
== See also ==
 
== See also ==

Latest revision as of 18:53, 30 June 2021

Problem

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: $\textbf{(A) } \qquad$ equal to the semi-circumference of the original circle $\textbf{(B) } \qquad$ equal to the diameter of the original circle $\textbf{(C) } \qquad$ greater than the diameter, but less than the semi-circumference of the original circle $\textbf{(D) } \qquad$ that is infinite $\textbf{(E) }$ greater than the semi-circumference

Solution

Note that the half the circumference of a circle with diameter $d$ is $\frac{\pi*d}{2}$.

Let's call the diameter of the circle D. Dividing the circle's diameter into n parts means that each semicircle has diameter $\frac{D}{n}$, and thus each semicircle measures $\frac{D*pi}{n*2}$. The total sum of those is $n*\frac{D*pi}{n*2}=\frac{D*pi}{2}$, and since that is the exact expression for the semi-circumference of the original circle, the answer is $\boxed{A}$.

See also

1952 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 42 		Followed by
Problem 44
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All AHSME Problems and Solutions

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