Three solid toys are in the shapes of a right circular cylinder, a hemisphere, and a right circular cone. All three solids have the same circular base, and the height of each toy is 2√2 cm. What is the ratio of the total surface areas of the cylinder, hemisphere, and cone respectively?

Introduction / Context:
This problem compares the total surface areas of three different solids: a right circular cylinder, a hemisphere, and a right circular cone. All three solids share the same circular base and are given the same height. The question tests understanding of surface area formulas and how changing shape (while keeping base and height fixed) affects total surface area.

Given Data / Assumptions:

• All three toys have the same circular base of radius r centimetres.
• The height of each toy is 2√2 cm.
• For the hemisphere, the height equals its radius, so r = 2√2 cm.
• Total surface area is taken as curved surface plus base area for each solid.
• We need the ratio of total surface areas of cylinder : hemisphere : cone.

Concept / Approach:
We use standard total surface area formulas for each solid in terms of the common radius r and the height h. Because r and h are the same in all three, the constant π and any common factors will cancel when we form the ratio. The key steps are to compute each total surface area in terms of π and simple numbers, then divide through by the common factor to simplify the ratio.

Step-by-Step Solution:
Step 1: For the cylinder, radius r = 2√2 cm and height h = 2√2 cm. Step 2: Total surface area of cylinder = 2 * π * r * h + 2 * π * r^2. Step 3: Compute r^2 = (2√2)^2 = 8. So cylinder area = 2π * (2√2) * (2√2) + 2π * 8 = 2π * 8 + 16π = 32π. Step 4: For the hemisphere of radius r, total surface area (curved plus base) = 3π * r^2 = 3π * 8 = 24π. Step 5: For the cone, radius r = 2√2 and height h = 2√2. Slant height l = √(r^2 + h^2) = √(8 + 8) = √16 = 4. Step 6: Total surface area of cone = π * r * l + π * r^2 = π * (2√2) * 4 + π * 8 = 8√2 π + 8π. Step 7: Form the ratio cylinder : hemisphere : cone = 32π : 24π : (8√2 π + 8π). Step 8: Divide every term by 8π to simplify: 32π / 8π = 4, 24π / 8π = 3, and (8√2 π + 8π) / 8π = √2 + 1. Step 9: So the required ratio is 4 : 3 : (√2 + 1).

Verification / Alternative check:
You can verify the ratio numerically by taking π as 3.14 and √2 as about 1.414, then calculating each area in square centimetres. When you simplify the three values, you will see they reduce to approximately the same ratio 4 : 3 : (√2 + 1). Because all shapes share the same base and height, no hidden scaling factors are present, so the ratio is stable regardless of the numerical value chosen for π.

Why Other Options Are Wrong:
Option b, 4 : 3 : (2 + √2), would correspond to a cone area that is too large by an extra 1 in the last term. Option c, 4 : 3 : 2√2 ignores the base area contribution and does not match the total surface area expression. Option d, 2 : 1 : (1 + √2), does not match the relative sizes of cylinder and hemisphere derived from their formulas, since the cylinder area is clearly more than just double the hemisphere area for this radius and height combination.

Common Pitfalls:
A common mistake is to forget to include the base area when computing total surface area, especially for the cone and hemisphere. Another frequent error is to use the height instead of the slant height for the cone's curved surface area. Some learners also assume r is an unknown rather than recognising that r = 2√2 for the hemisphere based on its height, which can lead to inconsistent expressions. Carefully using each formula and simplifying step by step avoids these issues.

Final Answer:
The ratio of total surface areas of the cylinder, hemisphere, and cone is 4 : 3 : (√2 + 1).