A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. If a right circular cylinder circumscribes the toy, find the difference of the volumes of the cylinder and the toy. (Take )
Answer: The volume of the toy is . The difference between the volumes of the cylinder and the toy is .
Step-by-step solution
Step 1: Calculate the radius of the base
The problem provides the diameter of the base of the cone. Since a hemisphere is surmounted by the cone, the radius of the hemispherical part is the same as the radius of the cone's base. We calculate the radius by dividing the diameter by 2.
Step 2: Calculate the volume of the cone
The formula for the volume of a right circular cone is given by one-third of the product of pi, the square of the radius of the base, and its height. We substitute the known values for the radius and height of the cone.
Step 3: Calculate the volume of the hemisphere
The formula for the volume of a hemisphere is two-thirds of the product of pi and the cube of its radius. We use the calculated radius for the hemisphere.
Step 4: Calculate the total volume of the toy
The total volume of the toy is the sum of the volume of the cone and the volume of the hemisphere, as the toy is formed by these two parts combined.
Step 5: Determine the dimensions of the circumscribing cylinder
When a right circular cylinder circumscribes the toy, its radius will be equal to the radius of the toy's base. The total height of the cylinder will be the sum of the height of the cone and the radius of the hemisphere, as the hemisphere's height is equal to its radius.
Step 6: Calculate the volume of the cylinder
The formula for the volume of a right circular cylinder is the product of pi, the square of its radius, and its height. We use the dimensions determined in the previous step.
Step 7: Calculate the difference in volumes
To find the difference between the volumes of the cylinder and the toy, we subtract the calculated volume of the toy from the calculated volume of the cylinder.
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