The curved surface area of a right circular cylinder is 1386 square centimetres and its height is 21 cm. Taking π = 22/7, what is the radius (in centimetres) of the cylinder?

Introduction / Context:
This question involves finding the radius of a right circular cylinder when its curved (lateral) surface area and height are given. The curved surface area of a cylinder is directly related to its radius and height through a standard formula. By substituting the given values and using the provided value of π, we can algebraically solve for the radius. This tests formula recall and basic manipulation of equations.

Given Data / Assumptions:
- Curved surface area (CSA) of the cylinder = 1386 square centimetres.- Height h = 21 cm.- π is given as 22/7.- Formula for curved surface area of a cylinder: CSA = 2 * π * r * h.- We need to find the radius r in centimetres.

Concept / Approach:
The curved surface area of a cylinder is the area of the side surface, which can be thought of as a rectangle with one side equal to the height and the other side equal to the circumference of the base. The formula CSA = 2 * π * r * h comes from this interpretation. To find r, we rearrange the formula to r = CSA / (2 * π * h). Substituting the given values into this rearranged formula and simplifying will yield the radius.

Step-by-Step Solution:
Step 1: Start with the formula: CSA = 2 * π * r * h.Step 2: Given CSA = 1386 square centimetres, h = 21 cm, and π = 22/7.Step 3: Substitute the known values: 1386 = 2 * (22/7) * r * 21.Step 4: Simplify the right hand side: 2 * (22/7) * 21 * r.Step 5: Note that 21 / 7 = 3, so (22/7) * 21 = 22 * 3 = 66.Step 6: Therefore 2 * (22/7) * 21 = 2 * 66 = 132.Step 7: The equation becomes 1386 = 132 * r.Step 8: Solve for r: r = 1386 / 132.Step 9: Simplify the fraction: 1386 / 132 = 10.5.Step 10: Therefore, r = 10.5 cm.

Verification / Alternative check:
To verify, substitute r = 10.5 and h = 21 back into the CSA formula. CSA = 2 * π * r * h = 2 * (22/7) * 10.5 * 21. Compute 10.5 = 21 / 2, so r = 21 / 2. Then 2 * (22/7) * (21 / 2) * 21 = (2 and 2 cancel) ⇒ (22/7) * 21 * 21. Now (22/7) * 21 = 66, and 66 * 21 = 1386. This exactly matches the given curved surface area, confirming that r = 10.5 cm is correct.

Why Other Options Are Wrong:
- 21 cms: This would give CSA = 2 * (22/7) * 21 * 21 = 2772, which is double the given value.- 5.25 cms: This is half of 10.5; substituting it would give CSA = 693, which is half of 1386.- 15.75 cms: This would produce a CSA much larger than 1386 when used in the formula.- 7 cms: Substituting r = 7 gives CSA = 2 * (22/7) * 7 * 21 = 924, which is not equal to 1386.

Common Pitfalls:
Some students may mistakenly use the total surface area formula (2 * π * r * h + 2 * π * r^2) instead of just the curved surface area formula. Others might perform algebraic steps incorrectly, especially when simplifying fractions with π = 22/7. Careful cancellation of factors and methodical substitution are essential to avoid miscalculations.

Final Answer:
The radius of the cylinder is 10.5 centimetres.