Three balls are packaged in a cylindrical container as shown below. The balls just touch the top, bottom, and sides of the cylinder. The diameter of each ball is 13 cm.
a. What is the volume of the cylinder rounded to the nearest cubed centimeter?
b. What is the total volume of the three balls rounded to the nearest cubed centimeter?
c. What percent of the volume of the container is occupied by the three balls.
a. The diameter of each ball is 13 cm, so the radius is 6.5 cm. Therefore, the height of the cylinder is also 13 cm (since the balls just touch the top and bottom). The formula for the volume of a cylinder is V = πr^2h, so substituting in the values gives:
V = π(6.5 cm)^2(13 cm)
V ≈ 3558 cm^3
Rounding to the nearest cubed centimeter:
V ≈ 3558 cm^3
b. The formula for the volume of a sphere (a ball is just a sphere in three dimensions) is V = (4/3)πr^3. Substituting in the radius of 6.5 cm gives:
V = (4/3)π(6.5 cm)^3
V ≈ 1796 cm^3 (rounded to the nearest cubed centimeter)
Therefore, the total volume of the three balls is:
3(1796 cm^3) = 5388 cm^3 (rounded to the nearest cubed centimeter)
c. The volume of the container is 3558 cm^3 (from part a), and the volume of the three balls is 5388 cm^3 (from part b). Therefore, the percentage of the volume of the container occupied by the three balls is:
(5388 cm^3 / 3558 cm^3) × 100% ≈ 151%
This answer doesn't make sense - it suggests that the balls are taking up more space than the container itself! This is likely due to rounding errors. In reality, the balls should take up less than 100% of the container's volume.