A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 8 cm. Determine the volume of the toy. If a cube circumscribes the toy, then find the difference of the volumes of the cube and the toy. Also, find the total surface area of the toy
Solution:
Given, a solid toy is in the form of a hemisphere surmounted by a right circular cone.
The height and diameter of the base of the cone is 4 cm and 8 cm.
We have to find the volume of the toy.
A cube circumscribes the toy.
We have to find the difference of the volumes of the cube and the toy and total surface area of the toy.
Diameter of base of cone = diameter of hemisphere = 8 cm
Radius = 8/2 = 4 cm
Volume of hemisphere = (2/3)πr³
= (2/3)(22/7)(4)³
= 134.095 cm³
Volume of the cone = (1/3)πr²h
= (1/3)(22/7)(4)²(4)
= 67.047 cm³
Volume of the toy = volume of hemisphere + volume of cone
= 134.095 + 67.047
= 201.142 cm³
Therefore, the volume of the toy is 201.42 cm³
Given, a cube circumscribes the toy
So, the edge of the cube = diameter of the hemisphere = 8 cm
Volume of cube = a³
= (8)³
= 512 cm³
Difference in volume of cube and toy = volume of cube - volume of toy
= 512 - 201.142
= 310.858 cm³
Total surface area of the toy = curved surface area of the cone + curved surface area of hemisphere.
Curved surface area of the cone = πrl
Slant height, l = √h² + r²
= √4² + 4²
= √32
= 5.657 cm
Curved surface area of the cone = (22/7)(4)(5.657)
= 71.117 cm²
Curved surface area of hemisphere = 2πr²
= 2(22/7)(4)²
= (44/7)(16)
= 100.571 cm²
Total surface area of the toy = 71.117 + 100.571
= 171.688 cm²
Therefore, the total surface area of the toy is 171.688 cm².
✦ Try This: A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 7 cm and the diameter of the base is 10 cm. Determine the volume of the toy. If a cube circumscribes the toy, then find the difference of the volumes of the cube and the toy. Also, find the total surface area of the toy.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 13
NCERT Exemplar Class 10 Maths Exercise 12.4 Sample Problem 2
A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 8 cm. Determine the volume of the toy. If a cube circumscribes the toy, then find the difference of the volumes of the cube and the toy. Also, find the total surface area of the toy
Summary:
A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 8 cm. The volume of the toy is 201.42 cm³. If a cube circumscribes the toy, then the difference of the volumes of the cube and the toy is 310.858 cm³. Also, the total surface area of the toy is 171.688 cm²
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