Question

The internal and external radii of a hollow hemisphere are \(5\sqrt{2} \text{ cm}\) and \(10 \text{ cm}\) respectively. A cone of height \(5\sqrt{7} \text{ cm}\) and radius \(5\sqrt{2} \text{ cm}\) is surmounted on the hemisphere as shown in the figure. Find the total surface area of the object in terms of \(\pi\). (Use \(\sqrt{2} = 1.4\))

Hint:

For hollow objects, "total surface area" includes both internal and external visible surfaces unless it's explicitly described as a closed solid. Don't forget the flat ring connecting the inner and outer shells.

Answer 1

Step 1: Calculate Slant Height of Cone
Given:
h = 5√7 cm
r = 5√2 cm

l = √(h² + r²)
= √[(5√7)² + (5√2)²]
= √[175 + 50]
= √225
= 15 cm

Step 2: Curved Surface Area of Cone
CSA(cone) = π r l
= π × 5√2 × 15
= 75√2 π cm²

Step 3: Area of Top Ring
External radius R = 10 cm
Internal radius r = 5√2 cm

Area of ring = π(R² − r²)
= π(100 − 50)
= 50π cm²

Step 4: Curved Surface Area of Hemisphere
Outer CSA = 2πR²
= 2π(100)
= 200π cm²

Inner CSA = 2πr²
= 2π(50)
= 100π cm²

Step 5: Total Surface Area
TSA = 75√2π + 50π + 200π + 100π
= 75√2π + 350π

If √2 ≈ 1.4,
75 × 1.4 = 105

So,
TSA ≈ 105π + 350π
= 455π cm²

Final Answer:
Total Surface Area = 455π cm²