The surface area of a sphere is 154 square centimetres. Using π = 22/7, what is the diameter (in centimetres) of the sphere?

Introduction / Context:
This question concerns a sphere for which the total surface area is given. The task is to find the diameter of the sphere. We use the standard surface area formula for a sphere and the given numerical value of π to back calculate the radius, and then the diameter. This is a straightforward but important application of formula manipulation in geometry.

Given Data / Assumptions:
- Surface area of the sphere S = 154 square centimetres.- π is given as 22/7.- Formula for surface area of a sphere: S = 4 * π * r^2.- We need to find the diameter, which is 2 * r.

Concept / Approach:
The total surface area S of a sphere depends on the square of its radius r through the formula S = 4 * π * r^2. Hence, if S and π are known, we can solve for r^2 by rearranging the formula to r^2 = S / (4 * π). After calculating r^2, we find r by taking the square root. The diameter is then simply 2 * r. Careful substitution and simplification using the given π value leads to the correct answer.

Step-by-Step Solution:
Step 1: Start with the formula S = 4 * π * r^2.Step 2: Substitute S = 154 and π = 22/7.Step 3: The equation becomes 154 = 4 * (22/7) * r^2.Step 4: Compute 4 * (22/7) = 88 / 7.Step 5: So 154 = (88 / 7) * r^2.Step 6: Multiply both sides by 7 to clear the fraction: 154 * 7 = 88 * r^2.Step 7: 154 * 7 = 1078.Step 8: Thus, 1078 = 88 * r^2.Step 9: Solve for r^2: r^2 = 1078 / 88.Step 10: Simplify 1078 / 88 = 12.25.Step 11: Therefore, r = √12.25 = 3.5 cm.Step 12: The diameter d = 2 * r = 2 * 3.5 = 7 cm.

Verification / Alternative check:
We can check by substituting r = 3.5 back into the surface area formula. r^2 = 12.25. Then S = 4 * π * r^2 = 4 * (22/7) * 12.25. Compute 4 * (22/7) = 88 / 7. Now S = (88 / 7) * 12.25. Since 12.25 = 49 / 4, we have S = (88 / 7) * (49 / 4) = (88 * 49) / 28. Simplifying (88 * 49) / 28 gives 154, exactly the given surface area, confirming that r = 3.5 cm and d = 7 cm are correct.

Why Other Options Are Wrong:
- 3.5: This is the radius, not the diameter, so it is only half of the required answer.- 14: This would correspond to a radius of 7 cm, giving a much larger surface area than 154 square centimetres.- 10.5: This would require r = 5.25 cm, and the surface area would be significantly larger than 154.- 5.25: Again this is neither correct for the radius nor diameter and gives an incorrect surface area when substituted.

Common Pitfalls:
Many learners forget the factor of 4 in the sphere surface area formula and mistakenly use S = π * r^2, which is the formula for the area of a circle, not a sphere. Others may compute the radius correctly but forget to double it when asked for the diameter. Keeping track of whether the question asks for radius or diameter is crucial to providing the correct final value.

Final Answer:
The diameter of the sphere is 7 centimetres.