The ratio of the radii of two circles is $4: 5$. What is the ratio of the circumferences of the two circles?
A. $16: 25$
B. $\pi: 5$
C. $4: 5$
D. $4: 5$
Answer (2)
The ratio of the circumferences of two circles, given the ratio of their radii as 4:5, is also 4:5. Thus, the correct answer is option C. This is due to the direct proportionality of the circumferences to their respective radii.
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Define the radii r 1 and r 2 of the two circles and express their ratio as r 2 r 1 = 5 4 .
Use the formula for the circumference of a circle, C = 2 π r , to find the circumferences C 1 = 2 π r 1 and C 2 = 2 π r 2 .
Determine the ratio of the circumferences: C 2 C 1 = 2 π r 2 2 π r 1 = r 2 r 1 .
Substitute the given ratio of the radii to find the ratio of the circumferences: C 2 C 1 = 5 4 , so the ratio is 4 : 5 .
Explanation
Define radii and circumferences Let r 1 and r 2 be the radii of the two circles, and let C 1 and C 2 be their respective circumferences. We are given that the ratio of the radii is 4 : 5 , which means:
Express the ratio of radii r 2 r 1 = 5 4
State the circumference formula The circumference of a circle is given by the formula C = 2 π r , where r is the radius. Therefore, the circumferences of the two circles are:
Circumference of the first circle C 1 = 2 π r 1
Circumference of the second circle C 2 = 2 π r 2
Find the ratio of circumferences We want to find the ratio of the circumferences, which is:
Express the ratio of circumferences C 2 C 1 = 2 π r 2 2 π r 1
Simplify the ratio We can simplify this ratio by canceling out the common factor of 2 π :
Simplified ratio of circumferences C 2 C 1 = r 2 r 1
Substitute the given ratio Since we know that r 2 r 1 = 5 4 , we can substitute this into the ratio of the circumferences:
Final ratio of circumferences C 2 C 1 = 5 4
Conclusion Therefore, the ratio of the circumferences of the two circles is 4 : 5 .
Examples
Understanding the ratio of circumferences is useful in various real-world applications. For example, when designing gears or pulleys, the ratio of their circumferences determines the speed and torque transmission. If one gear has a circumference twice as large as another, it will rotate half as fast, demonstrating an inverse relationship between circumference and rotational speed. This principle is crucial in mechanical engineering for creating efficient and reliable systems.
