A solid sphere of diameter 7 cm is cut into two equal hemispheres. By how many square centimetres does the total surface area increase?

Introduction / Context:
This question compares the total surface area of a solid sphere with that of two hemispheres obtained by cutting the sphere into two equal halves. Cutting a sphere exposes additional flat circular surfaces, increasing the total surface area. Understanding how surface areas change when solids are cut or recombined is important in geometry and practical applications such as material coating or heat transfer calculations.

Given Data / Assumptions:
• The original solid is a sphere with diameter 7 cm, so its radius r = 3.5 cm.
• The sphere is cut into two equal hemispheres.
• Surface area of a sphere is 4πr².
• Surface area of a hemisphere includes both the curved surface and the flat circular base.
• We need the increase in total surface area after cutting, in square centimetres.

Concept / Approach:
First compute the total surface area of the original sphere using 4πr². Then consider each hemisphere. A hemisphere has curved surface area equal to half of the full sphere surface, that is 2πr², plus the area of the base circle, which is πr². So the surface area of one hemisphere is 3πr². For two hemispheres, multiply by 2 to get 6πr². The increase in total surface area is the difference between 6πr² and 4πr², which simplifies to 2πr². Substituting r = 3.5 cm and a standard approximation for π gives the numerical answer.

Step-by-Step Solution:
Step 1: Radius of the sphere r = diameter / 2 = 7 / 2 = 3.5 cm. Step 2: Total surface area of the original sphere is S_sphere = 4πr². Step 3: Compute r² = 3.5² = 12.25, so S_sphere = 4π * 12.25 = 49π square centimetres. Step 4: Surface area of one hemisphere is S_hemi = curved area + base area = 2πr² + πr² = 3πr². Step 5: For two hemispheres, the total surface area is S_two = 2 * 3πr² = 6πr² = 6π * 12.25 = 73.5π square centimetres. Step 6: The increase in surface area is ΔS = S_two - S_sphere = 73.5π - 49π = 24.5π. Step 7: Take π = 22/7. Then ΔS = 24.5 * 22 / 7. Since 24.5 = 49 / 2, we get ΔS = (49 / 2) * 22 / 7 = (49 * 22) / (14) = (49 / 14) * 22 = 3.5 * 22 = 77 square centimetres.

Verification / Alternative check:
We can check the computations directly. Using r = 3.5 cm and π = 22/7, S_sphere = 4πr² = 4 * 22/7 * 12.25. Since 12.25 = 49 / 4, this becomes 4 * 22/7 * 49 / 4 = 22 * 7 = 154 square centimetres. For two hemispheres, S_two = 6πr² = 6 * 22/7 * 12.25, which simplifies to 6 * 22/7 * 49/4 = 3 * 22 * 7 / 7 = 3 * 22 * 7 / 4 when computed carefully, giving 231 square centimetres. The difference 231 - 154 equals 77 square centimetres, confirming the earlier result.

Why Other Options Are Wrong:
154 represents the total surface area of the original sphere, not the increase in area.
38.5 and 87 are smaller values that do not equal the exact computed increase 24.5π when converted using π = 22/7.
96 does not match any clear combination of sphere and hemisphere surface areas. Only 77 square centimetres is the correct difference between the two total surface areas.

Common Pitfalls:
A common mistake is to forget to add the base circles when computing the surface area of hemispheres, leading to an underestimate. Another error is to subtract in the wrong order or to miscalculate r². Some learners also confuse diameter and radius when applying formulas. Keeping track of which surfaces are exposed before and after cutting, and using the formulas consistently, helps avoid these errors.

Final Answer:
The total surface area increases by 77 square centimetres when the sphere is cut into two hemispheres.