A wooden bowl is in the shape of a hollow hemisphere. The internal radius is 8 cm and the thickness of the wood is 1 cm. What is the total surface area of the bowl (in square centimetres)?

Introduction / Context:
This is another hollow hemispherical bowl problem, similar in structure to earlier ones but with different radii. It reinforces how to compute the total exposed surface area of a bowl that has both inner and outer curved surfaces and an exposed rim, using appropriate radii for the inner and outer surfaces and the ring at the top.

Given Data / Assumptions:

• Internal radius r = 8 cm.
• Thickness of the bowl = 1 cm, so external radius R = 8 + 1 = 9 cm.
• The bowl is hollow and open at the top.
• Total surface area includes inner curved surface, outer curved surface, and the ring shaped rim at the top.
• Use π ≈ 3.14 (or similar) to match the numerical options.

Concept / Approach:
The total surface area TSA of a hollow hemispherical bowl is found by adding:

• Inner curved surface area = 2πr^2.
• Outer curved surface area = 2πR^2.
• Rim (ring) area = π(R^2 − r^2).

Adding these gives TSA = 2πr^2 + 2πR^2 + π(R^2 − r^2) = π(r^2 + 3R^2). We substitute r = 8, R = 9 and then approximate numerically.

Step-by-Step Solution:
Step 1: Set inner radius r = 8 cm and outer radius R = 9 cm. Step 2: Compute r^2 = 8^2 = 64 and R^2 = 9^2 = 81. Step 3: Use the compact formula TSA = π(r^2 + 3R^2). Step 4: Substitute values: TSA = π(64 + 3 * 81) = π(64 + 243) = π * 307. Step 5: Using π ≈ 3.14, TSA ≈ 307 * 3.14. Step 6: Compute 307 * 3.14 ≈ 307 * (3 + 0.14) ≈ 921 + 42.98 ≈ 963.98 square centimetres. Step 7: With slightly more precise π ≈ 3.1416, TSA is around 964.8 square centimetres, very close to the option 964.85 sq cm.

Verification / Alternative check:
You can also compute each component separately:

• Inner curved area = 2πr^2 = 2π * 64 = 128π.
• Outer curved area = 2πR^2 = 2π * 81 = 162π.
• Rim area = π(R^2 − r^2) = π(81 − 64) = 17π.

Total = (128 + 162 + 17)π = 307π, consistent with the compact formula. Using π ≈ 3.1416 again yields approximately 964 square centimetres, matching the chosen option.

Why Other Options Are Wrong:
281.29 sq cm is much too small and does not even match a single curved surface area. 1125.14 sq cm and 1687.71 sq cm are much larger than the calculated value and reflect miscounting surfaces or using the wrong radii. Only 964.85 sq cm lines up well with the accurate calculation of 307π square centimetres.

Common Pitfalls:
As with similar problems, students may forget to include the rim area or mistakenly use only one radius for all surfaces. Another error is to confuse volume and surface area formulas or to treat the hemisphere as a full sphere. Being careful to identify all exposed surfaces and applying the correct radii ensures the correct total surface area.

Final Answer:
The total surface area of the wooden bowl is approximately 964.85 square centimetres.