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Zorluk: OrtaCircles

Two circular dials on a control panel have a radius ratio of 2:32:3. If the circumference of the smaller dial is 12π centimeters12\pi \text{ centimeters}, what is the circumference of the larger dial?

  1. A
    8π cm8\pi \text{ cm}
  2. 18π cm18\pi \text{ cm}Cevap
  3. C
    36π cm36\pi \text{ cm}
  4. D
    81π cm81\pi \text{ cm}

Cevap

The circumference of the larger dial is 18π cm18\pi \text{ cm}.
The ratio of the radii is 2:3. The smaller dial has a circumference of 12π12\pi, meaning its radius is 66 (since C=2πrC = 2\pi r). Setting up the proportion 6r=23\frac{6}{r} = \frac{2}{3} gives a larger radius of 99. The circumference of the larger dial is then 2π(9)=18π cm2\pi(9) = 18\pi \text{ cm}.

Adım Adım Çözüm

1
Determine the radius of the smaller dial.
2πr=12π→r=6 cm2\pi r = 12\pi \rightarrow r = 6 \text{ cm}
The radius of the smaller dial is required to use the given ratio.
2
Apply the ratio to find the radius of the larger dial.
6rlarge=23→2rlarge=18→rlarge=9 cm\frac{6}{r_{large}} = \frac{2}{3} \rightarrow 2r_{large} = 18 \rightarrow r_{large} = 9 \text{ cm}
The problem states the radius ratio between the smaller and larger dials is 2:3.
3
Calculate the circumference of the larger dial.
C=2π(9)=18π cmC = 2\pi(9) = 18\pi \text{ cm}
The question asks for the circumference of the larger dial.

Anahtar Kavram

Using proportions to find unknown circle dimensions and calculating circumference.

Alternatif Yöntem

Because circumference is directly proportional to radius (C=2πrC = 2\pi r), the circumferences of two circles share the exact same ratio as their radii (2:32:3). Therefore, you can set up the proportion 12πC=23\frac{12\pi}{C} = \frac{2}{3} to solve directly for C=18πC = 18\pi without needing to calculate the specific radii.
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