Final answer:
The removed cone has dimensions that are proportional to the original cone, and when the radius is halved, the height is also halved. Both scale to give a volume that is 1/8 of the original cone using the formula V = 1/3πr²h. Option b) is correct because volume is proportional to the dimensions.
Step-by-step explanation:
The cone that is removed has a radius of 2cm and a height of 5cm because it is mathematically similar to the original cone. This means that all dimensions of the removed cone are scaled down by the same factor. To determine why these specific dimensions result in a volume that is 1/8 of the original cone, we must use the formula for the volume of a cone, which is V = 1/3πr²h.
Let's first find the volume of the original cone:
V_original = 1/3π(4 cm)²(10 cm) = 1/3π(16 cm²)(10 cm) = 160/3π cm³
To find the volume of the removed cone that is 1/8 of the original, divide this volume by 8:
V_removed = (160/3π cm³) / 8 = 20/3π cm³
Because the removed cone is similar to the original cone, if the radius of the original is halved, the height is also halved. Hence, the removed cone would have a radius of 2cm (half of 4cm) and a height of 5cm (half of 10cm). These dimensions result in a volume that is exactly 1/8 of the original cone:
V_removed = 1/3π(2 cm)²(5 cm) = 1/3π(4 cm²)(5 cm) = 20/3π cm³
Therefore, the removed cone's dimensions are directly proportional to its volume compared to the original cone (option b).