The total surface area of a solid hemisphere is 166.32 square centimetres. Using π ≈ 3.14, what is the radius (in centimetres) of the hemisphere?

Introduction / Context:
This question focuses on finding the radius of a hemisphere when its total surface area is given. The total surface area of a solid hemisphere includes both the curved (outer) surface and the flat circular base. Understanding and applying the correct surface area formula for a hemisphere and then solving for the radius using basic algebra are the key skills tested here.

Given Data / Assumptions:
- The solid shape is a hemisphere.- Total surface area (TSA) = 166.32 square centimetres.- π is approximately 3.14.- TSA of a hemisphere includes both curved surface and base.- Formula for total surface area of a hemisphere: TSA = 3 * π * r^2.

Concept / Approach:
For a sphere of radius r, surface area = 4 * π * r^2. For a hemisphere, the curved surface area alone is 2 * π * r^2. However, a solid hemisphere also has a flat circular base area π * r^2. Therefore, total surface area becomes TSA = 2 * π * r^2 + π * r^2 = 3 * π * r^2. We use the given TSA to set up an equation 3 * π * r^2 = 166.32, then solve for r^2 and finally for r. Since π is given as 3.14, we substitute this value into the formula.

Step-by-Step Solution:
Step 1: Write formula for total surface area of a hemisphere: TSA = 3 * π * r^2.Step 2: Given TSA = 166.32 square centimetres.Step 3: Substitute into the formula: 166.32 = 3 * π * r^2.Step 4: Take π ≈ 3.14.Step 5: Compute 3 * π = 3 * 3.14 = 9.42.Step 6: So the equation becomes 166.32 = 9.42 * r^2.Step 7: Solve for r^2: r^2 = 166.32 / 9.42.Step 8: Perform the division: r^2 ≈ 17.66 (approximately).Step 9: Take square root: r ≈ √17.66 ≈ 4.2 cm.Step 10: This matches the option 4.2 cm.

Verification / Alternative check:
We can check by substituting r = 4.2 back into TSA = 3 * π * r^2. First, r^2 = 4.2^2 = 17.64. Then 3 * π * r^2 ≈ 3 * 3.14 * 17.64. Compute 3 * 3.14 = 9.42, then 9.42 * 17.64 ≈ 166.1 to 166.3, which is very close to the given 166.32, and any slight difference is due to rounding π. This confirms that r ≈ 4.2 cm is correct within the rounding used in the problem.

Why Other Options Are Wrong:
- 8.4 cm: Doubling the radius would make r^2 four times larger, which would make TSA roughly four times larger than 166.32.- 1.4 cm: A radius this small would give r^2 = 1.96 and TSA ≈ 3 * 3.14 * 1.96 ≈ 18.5, which is far below 166.32.- 2.1 cm: This radius gives r^2 = 4.41 and TSA ≈ 3 * 3.14 * 4.41 ≈ 41.6, still much less than required.- 6.3 cm: This gives r^2 ≈ 39.69 and TSA ≈ 3 * 3.14 * 39.69 ≈ 373.8, which is more than double the given surface area.

Common Pitfalls:
Some students mistakenly use the formula for only the curved surface area (2 * π * r^2) instead of the total surface area for a hemisphere, omitting the base area. Others might use 4 * π * r^2, which is the formula for a full sphere. Incorrect substitution of π or premature rounding can also lead to inaccurate results. Always ensure the correct formula is used for the specific solid described in the question and apply the given value of π as instructed.

Final Answer:
The radius of the hemisphere is approximately 4.2 cm.