NCERT Solutions Class 10 Maths Chapter 12 — Surface Areas & Volumes

Solution:

Step 1: The total surface area of a combined solid is the sum of the curved (or exposed) surface areas of its individual components.

Step 2: When a cone is mounted on a hemisphere, the circular base of the cone and the circular top of the hemisphere are joined together, becoming internal surfaces.

Step 3: Therefore, the exposed surface area consists only of the curved surface area of the cone and the curved surface area of the hemisphere.

Solution:

Step 1: Volume is a measure of the three-dimensional space occupied by an object.

Step 2: When two or more solids are combined to form a new solid, the total volume of the resulting solid is the sum of the volumes of the individual solids, assuming there are no hollow spaces or overlaps beyond their joining surfaces.

Step 3: Surface area, on the other hand, measures the area of the exterior surfaces and is not directly additive in the same way.

Solution:

Step 1: The cubical block has 6 faces. When a hemisphere is placed on top of one face, a circular area on that face is covered by the base of the hemisphere.

Step 2: This covered area is no longer part of the exposed surface and must be excluded from the cube's total surface area.

Step 3: Therefore, the correct approach is: (TSA of Cube) - (Area of circular base of hemisphere) + (CSA of Hemisphere).

Solution:

Step 1: Given diameter = 4 cm, so radius (r) = 4/2 = 2 cm. Height of cone (h) = 6 cm.

Step 2: Volume of cone = (1/3) × π × r² × h = (1/3) × π × (2)² × 6 = (1/3) × π × 4 × 6 = 8π cm³.

Step 3: Volume of hemisphere = (2/3) × π × r³ = (2/3) × π × (2)³ = (2/3) × π × 8 = (16/3)π cm³.

Step 4: Total Volume = Volume of cone + Volume of hemisphere = 8π + (16/3)π = (24/3)π + (16/3)π = (40/3)π cm³ ≈ (40/3) × (22/7) = 880/21 ≈ 41.90 cm³.

Solution:

Step 1: Diameter = 7 m, so radius (r) = 7/2 m.

Step 2: Total length of the tank = 13.5 m. The height of the cylindrical part (h) = Total length - 2 × r = 13.5 - 2 × (7/2) = 13.5 - 7 = 6.5 m.

Step 3: Total Surface Area = CSA of cylinder + 2 × CSA of hemisphere.

Step 4: TSA = 2πrh + 2 × (2πr²) = 2πr(h + 2r) = 2 × (22/7) × (7/2) × (6.5 + 2 × 7/2) = 22 × (6.5 + 7) = 22 × 13.5 = 297 m².