A wooden bowl is in the shape of a hollow hemisphere with an internal radius of 7 cm and a uniform thickness of 1 cm. Using π = 22/7, what is the total surface area (in square centimetres) of the bowl (inside, outside, and rim)?

Introduction / Context:
This question deals with the surface area of a three dimensional solid, specifically a hollow hemisphere that is used as a wooden bowl. The bowl has an inner radius and a given thickness, so there is also an outer radius. The task is to find the total surface area, which includes the inner curved surface, the outer curved surface, and the circular rim at the open mouth of the bowl. Understanding the formulas for the surface areas of spheres and hemispheres is essential here.

Given Data / Assumptions:
- The bowl is a hollow hemisphere.- Internal radius r_in = 7 cm.- Thickness of the bowl = 1 cm.- Therefore, external radius r_out = 7 + 1 = 8 cm.- π is taken as 22/7.- Total surface area includes: inner curved surface area, outer curved surface area, and the area of the circular ring (rim) between the two radii.

Concept / Approach:
For a hemisphere, curved surface area = 2 * π * r^2. For a hollow hemisphere (bowl), there is an inner curved surface and an outer curved surface. In addition, the bowl has an exposed ring shaped surface at the mouth. The area of this ring is the difference between the areas of two circles: π * r_out^2 minus π * r_in^2. Therefore total surface area TSA is the sum of these three parts: 2 * π * r_in^2 + 2 * π * r_out^2 + π * (r_out^2 - r_in^2).

Step-by-Step Solution:
Step 1: Compute r_in^2 = 7^2 = 49.Step 2: Compute r_out^2 = 8^2 = 64.Step 3: Inner curved surface area = 2 * π * r_in^2 = 2 * π * 49.Step 4: Outer curved surface area = 2 * π * r_out^2 = 2 * π * 64.Step 5: Area of rim (ring) = π * (r_out^2 - r_in^2) = π * (64 - 49) = π * 15.Step 6: Total surface area TSA = 2 * π * 49 + 2 * π * 64 + π * 15.Step 7: Combine inside: 2 * 49 = 98, 2 * 64 = 128, so TSA = (98 + 128 + 15) * π = 241 * π.Step 8: Using π = 22/7, TSA = 241 * 22 / 7.Step 9: Calculate 241 * 22 = 5302. Then 5302 / 7 ≈ 757.43 square centimetres.

Verification / Alternative check:
We can check the reasonableness by comparing inner and outer curved surface areas. Inner curved area is 2 * π * 49, outer curved area is 2 * π * 64, which clearly makes the outer area larger. Adding a positive rim area increases the total further. Numerically these give approximately 2 * 3.14 * 49 ≈ 307.72 and 2 * 3.14 * 64 ≈ 401.92, and the rim is roughly 3.14 * 15 ≈ 47.1, which when summed is around 756.7. This is very close to 757.43 and confirms that the result is consistent.

Why Other Options Are Wrong:
- 710.29: This underestimates the total area and would correspond to omitting part of the surface, such as the rim or the difference between inner and outer areas.- 355.14: This is close to the order of a single curved surface only, not the total combined surfaces.- 908.91: This is too large, likely arising from double counting or using incorrect radii in the formula.- 531.42: This also underestimates the correct total and may result from including only inner or outer curved surface plus a small portion.

Common Pitfalls:
One common mistake is to consider only one curved surface area (either inner or outer) and ignore the other, which significantly changes the result. Another frequent error is neglecting the ring shaped rim at the open top, even though it is part of the exposed surface. Some students also confuse volume formulas with surface area formulas. Careful identification of every surface that is exposed to the air is crucial in such hollow solid problems.

Final Answer:
The total surface area of the wooden bowl is approximately 757.43 square centimetres.