The sum of the radius and height of a solid right circular cylinder is 20 cm. If its total surface area is 880 sq.cm, what is the volume of the cylinder in cubic centimetres?

Introduction / Context:
This question involves the geometry of a right circular cylinder. You are given a relationship between the radius and height and the total surface area. From these, you need to determine the volume. It tests your understanding of formulas for total surface area and volume of a cylinder and your ability to link them using algebra.

Given Data / Assumptions:
• Sum of radius and height is r + h = 20 cm.
• Total surface area of the cylinder is 880 sq.cm.
• The figure is a solid right circular cylinder with both circular ends included in the surface area.
• Use standard formulas with π left symbolic or approximated as 22/7 for final numerical value.

Concept / Approach:
For a cylinder of radius r and height h:
• Total surface area (TSA) = 2πrh + 2πr2 = 2πr(r + h).
• Volume V = πr2h.
We use the relation r + h = 20 to simplify the TSA expression. Once r is obtained from the surface area, h follows from r + h = 20. Then we compute the volume.

Step-by-Step Solution:
Step 1: TSA = 2πr(r + h). Given r + h = 20, TSA = 2πr × 20 = 40πr. Step 2: TSA is given as 880 sq.cm, so 40πr = 880. Step 3: Divide both sides by 40π: r = 880 / (40π). Step 4: Using π = 22/7, r = 880 / (40 × 22/7) = 880 × 7 / 880 = 7 cm. Step 5: From r + h = 20, we get h = 20 - 7 = 13 cm. Step 6: Volume V = πr2h = π × 72 × 13 = π × 49 × 13. Step 7: Compute V with π = 22/7: V = 49 × 13 × 22 / 7 = 7 × 13 × 22 = 2002 cm3.

Verification / Alternative check:
Recheck the TSA with r = 7 and h = 13. TSA = 2πrh + 2πr2 = 2π × 7 × 13 + 2π × 72 = 182π + 98π = 280π. With π = 22/7, TSA = 280 × 22 / 7 = 40 × 22 = 880 sq.cm, which matches the given value. This confirms that the dimensions are correct and so is the volume.

Why Other Options Are Wrong:
• 1760 cm3 comes from an incorrect combination of r and h or using only curved surface area.
• 8800 cm3 is far too large for these dimensions and would require much bigger radius or height.
• 4804 cm3 does not correspond to any consistent solution of the given equations.

Common Pitfalls:
Students sometimes forget that total surface area of a closed cylinder includes both circular ends, not just the curved surface. Another common mistake is to treat r + h as r × h. Careful reading and correct substitution into TSA = 2πr(r + h) are essential to avoid algebraic errors.

Final Answer:
The volume of the cylinder is 2002 cm3.