Multiple choice

The radius of the outer circumference of a cylindrical ring is $12 cm$ and the diameter of the cross-section is $1.5 cm$. Then, the volume of circular ring is

  1. $22.5\, cm^3$
  2. $125\, cm^3$
  3. $226\, cm^3$
  4. $1250\, cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The volume of a circular ring (torus) is given by the formula V = 2 * pi^2 * R * r^2, where R is the mean radius of the ring and r is the radius of the cross-section. Here, the cross-section radius is r = 1.5 / 2 = 0.75 cm, and the mean radius is R = 12 - 0.75 = 11.25 cm. Substituting these values gives V = 2 * pi^2 * 11.25 * 0.75^2, which is approximately 125 cubic cm.

AI explanation

Using Pappus's Centroid Theorem, the volume of the cylindrical ring is found by multiplying the cross-sectional area by the path length of its center. The radius of the cross-section is 1.5 divided by 2, equaling 0.75 cm, making the cross-sectional area 22 divided by 7 multiplied by the square of 0.75, which simplifies to 99 divided by 56 square centimeters. The mean radius of the ring is 12 minus 0.75, which equals 11.25 cm, making the path length 2 times pi times 11.25, or 45 times pi cm. Multiplying the area by the path length gives 99 divided by 56 multiplied by 45 times 22 divided by 7, resulting in a volume of approximately 125 cubic centimeters.

Keep practicing — related questions