Q1.
Two cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
Solution:
Volume of each cube = 64 cm³
⇒ side = ∛64 = 4 cm
Dimensions of cuboid = 8 cm × 4 cm × 4 cm
Surface area = 2(lb + bh + hl)
= 2(8×4 + 4×4 + 8×4)
= 2(32 + 16 + 32)
= 2×80 = 160 cm²
Q2.
A vessel is a hollow hemisphere mounted on a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Solution:
Radius (r) = 7 cm
Height of cylinder (h) = 13 − 7 = 6 cm
Inner surface area = 2πrh + 2πr² + πr²
= 2πrh + 3πr²
= 2×(22/7)×7×6 + 3×(22/7)×7×7
= 264 + 462 = 726 cm²
Q3.
A toy is in the form of a cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the total height of the toy is 15.5 cm. Find the total surface area of the toy.
Solution:
r = 3.5 cm, h = 15.5 − 3.5 = 12 cm
Slant height (l) = √(h² + r²) = √(12² + 3.5²) = 12.5 cm
Total surface area = πrl + 2πr²
= (22/7)×3.5×12.5 + 2×(22/7)×3.5×3.5
= 137.5 + 77 = 214.5 cm²
Q4.
A cubical block of side 7 cm is surmounted by a hemisphere. Find the greatest diameter of the hemisphere and the surface area of the solid.
Solution:
Greatest diameter = side of cube = 7 cm
Radius r = 3.5 cm
Surface area = 6a² − a² + 2πr²
= 5a² + 2πr²
= 5×7² + 2×(22/7)×3.5²
= 245 + 77 = 322 cm²
Q5.
A hemispherical depression is cut out from one face of a cubical block of edge l, such that the diameter of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
Solution:
Edge = a = l
Radius = a/2
Surface area = 6a² − πr² + 2πr²
= 6a² + πr²
= 6a² + π(a²/4)
= a²(6 + π/4)
∴ Surface area = 6a² + (πa² / 4)
Q6.
A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter is 5 mm. Find its surface area.
Solution:
r = 2.5 mm, h = 14 − 2r = 9 mm
Surface area = 2πrh + 4πr²
= 2×(22/7)×2.5×9 + 4×(22/7)×2.5²
= (990/7) + (550/7)
= (1540/7) = 220 mm²
Q7.
A tent is in the shape of a cylinder surmounted by a conical top. The height and diameter of the cylindrical part are 2.1 m and 4 m respectively. The slant height of the top is 2.8 m. Find the area of the canvas required to make the tent, excluding the base. Also, find the cost of the canvas at ₹500 per m².
Solution:
r = 2 m, h = 2.1 m, l = 2.8 m
Area = 2πrh + πrl
= 2π×2×2.1 + π×2×2.8
= 8.4π + 5.6π = 14π
= 14×3.14 = 43.96 ≈ 44 m²
Cost = 44×500 = ₹22,000
Q8.
From a solid cylinder (height 2.4 cm, diameter 1.4 cm), a conical cavity of the same height and diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm².
Solution:
r = 0.7 cm, h = 2.4 cm
l = √(r² + h²) = √(0.7² + 2.4²) = 2.5 cm
Total surface area = 2πrh + πrl + πr²
= 2×(22/7)×0.7×2.4 + (22/7)×0.7×2.5 + (22/7)×0.7²
= 3.36π + 1.75π + 0.49π = 5.6π
= 5.6×3.14 = 17.6 ≈ 18 cm²
Q9.
A wooden article was made by scooping out a hemisphere from each end of a solid cylinder whose height is 10 cm and base radius 3.5 cm. Find the total surface area of the article.
Solution:
r = 3.5 cm, h = 10 cm
Total surface area = 2πrh + 4πr²
= 2×(22/7)×3.5×10 + 4×(22/7)×3.5²
= 70π + 49π = 119π
= 119×(22/7) = 374 cm²