A toy is in the shape of a cone mounted on a hemisphere, both of radius 7 cm. The total height of the toy is 21 cm. Find the volume of the toy. (Use π = 22/7.)
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Step-by-Step Solution
Step 1: Identify Given Values and Calculate Cone Height
We are given that the radius of both the cone and the hemisphere is R=7 cm. The total height of the toy is H=21 cm. Since the hemisphere has a radius of 7 cm, its height is also 7 cm. To find the height of the cone (h), we subtract the height of the hemisphere from the total height of the toy.
Step 2: Calculate Height of the Cone
Substituting the given values, the height of the cone is 21 cm−7 cm=14 cm.
Step 3: Calculate Volume of Hemisphere
The formula for the volume of a hemisphere is 32πR3. We will use this formula to calculate the volume of the hemispherical part of the toy.
Step 4: Substitute Values for Hemisphere Volume
Substitute R=7 cm and π=722 into the hemisphere volume formula. After calculation, the volume of the hemisphere is 32156 cm3.
Step 5: Calculate Volume of Cone
The formula for the volume of a cone is 31πR2h. We will use this formula to calculate the volume of the conical part of the toy.
Step 6: Substitute Values for Cone Volume
Substitute R=7 cm, h=14 cm, and π=722 into the cone volume formula. After calculation, the volume of the cone is 32156 cm3.
Step 7: Calculate Total Volume of Toy
The total volume of the toy is the sum of the volume of the hemisphere and the volume of the cone. Adding the two volumes, we get 32156+32156=34312 cm3, which is approximately 1437.33 cm3.