Difference between revisions of "1952 AHSME Problems/Problem 43"
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== Problem == | == Problem == | ||
− | + | The diameter of a circle is divided into <math>n</math> equal parts. On each part a semicircle is constructed. As <math>n</math> becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length: | |
− | <math>\textbf{(A) } | + | <math>\textbf{(A) } \qquad</math> equal to the semi-circumference of the original circle |
− | \textbf{(B) } | + | <math>\textbf{(B) } \qquad</math> equal to the diameter of the original circle |
− | \textbf{(C) } | + | <math>\textbf{(C) } \qquad</math> greater than the diameter, but less than the semi-circumference of the original circle |
− | \textbf{(D) } | + | <math>\textbf{(D) } \qquad</math> that is infinite |
− | \textbf{(E) } | + | <math>\textbf{(E) } </math> greater than the semi-circumference |
== Solution == | == Solution == | ||
− | <math>\ | + | Note that the half the circumference of a circle with diameter <math>d</math> is <math>\frac{\pi*d}{2}</math>. |
+ | |||
+ | 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>. | ||
== See also == | == See also == |
Latest revision as of 11:28, 29 December 2024
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:
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
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
.
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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