A right circular cylinder has height 18 cm and radius 7 cm. The cylinder is cut into three equal smaller cylinders by two cuts parallel to the base, so each smaller cylinder has height 6 cm. What is the percentage increase in the total surface area after cutting?

Introduction / Context:
This mensuration question compares the surface area of a single right circular cylinder with the surface area after it is cut into three equal smaller cylinders. Although the total volume remains constant, cutting introduces new circular surfaces at the cross sections, which increases total surface area. The task is to compute this increase as a percentage.

Given Data / Assumptions:

• Original cylinder radius r = 7 cm.
• Original cylinder height H = 18 cm.
• Two cuts parallel to the base divide it into three cylinders each of height 6 cm.
• All pieces remain solid with no loss of material.
• We need the percentage increase in total surface area.

Concept / Approach:
Total surface area of a cylinder equals curved surface area plus the areas of the two circular bases. For radius r and height h, TSA = 2 * pi * r * h + 2 * pi * r^2. Initially there is one cylinder. After cutting, there are three cylinders with the same radius but smaller height. Each small cylinder has its own two circular bases, so more base surfaces appear. Curved surface area is proportional to height, so the total curved surface area of the three cylinders remains the same as that of the original cylinder, but the number of circular base areas increases from two to six.

Step-by-Step Solution:
Step 1: Original cylinder: radius r = 7 cm, height H = 18 cm.Step 2: Original TSA = 2 * pi * r * H + 2 * pi * r^2 = 2 * pi * 7 * 18 + 2 * pi * 7^2 = 252pi + 98pi = 350pi.Step 3: After cutting, each small cylinder has height h = 6 cm and radius 7 cm.Step 4: TSA of one small cylinder = 2 * pi * r * h + 2 * pi * r^2 = 2 * pi * 7 * 6 + 2 * pi * 49 = 84pi + 98pi = 182pi.Step 5: There are three such cylinders, so total TSA after cutting = 3 * 182pi = 546pi.Step 6: Increase in surface area = 546pi − 350pi = 196pi.Step 7: Percentage increase = (196pi / 350pi) * 100 = (196 / 350) * 100 = 56 percent.

Verification / Alternative check:
Observe that the curved surface area of the original cylinder is 2 * pi * r * H = 252pi. After cutting, combined curved surface area of three smaller cylinders is 3 * 2 * pi * r * 6 = 3 * 84pi = 252pi, which is unchanged. The only change is in the number of circular bases: initially there were 2 bases, now there are 6, so effectively 4 new base areas of pi * r^2 each are added. That extra area is 4 * pi * 7^2 = 196pi, confirming the earlier computation and simplifying the reasoning for percentage increase.

Why Other Options Are Wrong:

• 62, 48, and 52 percent do not match the exact calculation of 196π over 350π.
• 36 percent significantly underestimates the impact of adding four extra base circles.

Common Pitfalls:

• Forgetting that curved surface area stays the same while only the number of circular bases increases.
• Computing TSA for a single small cylinder and forgetting to multiply by three.
• Making arithmetic mistakes when simplifying 196 / 350 to a percentage.

Final Answer:
The total surface area increases by 56 percent after cutting the cylinder into three equal parts.