One circle has a circumference of 12 cm. Another circle has a circumference of 32 cm. What is the ratio of the radius of the smaller circle to the radius of the larger circle?
A. 3:8
B. 8:3
C. 1:3
D. 3:1
Answer (2)
The ratio of the radius of the smaller circle to the radius of the larger circle is 3:8. This is calculated using the circumferences of the circles and solving for their radii. Therefore, the answer is option A. 3:8.
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Define the circumference of the smaller circle as C 1 = 12 cm and express its radius as r 1 = π 6 .
Define the circumference of the larger circle as C 2 = 32 cm and express its radius as r 2 = π 16 .
Calculate the ratio of the radii as r 2 r 1 = π 16 π 6 .
Simplify the ratio to find the final answer: 8 3 .
Explanation
Define the circumference and radius of the smaller circle. Let C 1 be the circumference of the smaller circle and r 1 be its radius. We know that the circumference of a circle is given by the formula C = 2 π r , where C is the circumference and r is the radius. Therefore, for the smaller circle, we have:
Write the equation for the smaller circle's circumference. C 1 = 2 π r 1 = 12
Define the circumference and radius of the larger circle. Let C 2 be the circumference of the larger circle and r 2 be its radius. Similarly, for the larger circle, we have:
Write the equation for the larger circle's circumference. C 2 = 2 π r 2 = 32
State the objective. We want to find the ratio of the radius of the smaller circle to the radius of the larger circle, which is r 2 r 1 .
Solve for the radius of the smaller circle. From the equation for the smaller circle, we can express r 1 in terms of C 1 :
r 1 = 2 π C 1 = 2 π 12 = π 6
Solve for the radius of the larger circle. From the equation for the larger circle, we can express r 2 in terms of C 2 :
r 2 = 2 π C 2 = 2 π 32 = π 16
Calculate the ratio of the radii. Now, we can find the ratio r 2 r 1 :
r 2 r 1 = π 16 π 6 = π 6 ⋅ 16 π = 16 6 = 8 3
State the final answer. Therefore, the ratio of the radius of the smaller circle to the radius of the larger circle is 8 3 , which can be written as 3:8.
Examples
Understanding the ratio of circle radii is useful in many real-world applications. For example, when designing gears or pulleys, the ratio of their radii determines the mechanical advantage. If you have two gears, one with a radius of 3 cm and another with a radius of 8 cm, the gear ratio is 3:8. This means that for every 8 rotations of the smaller gear, the larger gear rotates 3 times. This concept is crucial in designing efficient mechanical systems.
