A right circular cylinder has height 28 cm and base radius 14 cm. From each of its two plane ends, a hemispherical cavity of radius 7 cm is cut out. What is the total surface area, in square centimetres, of the remaining solid surface?

Introduction / Context:
This problem deals with a composite solid formed by a right circular cylinder from which hemispherical cavities are removed at both ends. The aim is to compute the total surface area of the remaining solid. This tests understanding of curved surface areas of cylinders and hemispheres, as well as how to handle annular regions created when a smaller circular region is removed from a larger one at the ends.

Given Data / Assumptions:

• Height of the cylinder H = 28 cm.
• Base radius of the cylinder R = 14 cm.
• Each hemispherical cavity has radius r = 7 cm.
• One such hemispherical cavity is cut from each of the two plane ends of the cylinder.
• The required surface area includes the outer curved surface of the cylinder, the flat annular regions at both ends, and the curved surfaces of both hemispherical cavities.
• Use pi approximately equal to 22 / 7 for numerical evaluation.

Concept / Approach:
The total surface area of the remaining solid is the sum of several parts:

• Curved surface area of the original cylinder: 2 * pi * R * H.
• Two flat annular regions on the ends after cutting out circles of radius r from discs of radius R, each with area pi * (R^2 - r^2).
• Curved surface areas of the two hemispherical cavities, each equal to 2 * pi * r^2.

The total surface area is then computed by summing these contributions and simplifying.

Step-by-Step Solution:
Step 1: Curved surface area of the cylinder: A_cyl = 2 * pi * R * H = 2 * pi * 14 * 28 = 784 * pi.Step 2: Area of one base disc is pi * R^2 = pi * 14^2 = 196 * pi.Step 3: Area of the circular hole for each hemisphere is pi * r^2 = pi * 7^2 = 49 * pi.Step 4: Area of one annular flat region is pi * (R^2 - r^2) = pi * (196 - 49) = 147 * pi. There are two such regions, so their total area is 2 * 147 * pi = 294 * pi.Step 5: Curved surface area of one hemisphere is 2 * pi * r^2 = 2 * pi * 7^2 = 98 * pi. For two hemispherical cavities, total area is 2 * 98 * pi = 196 * pi.Step 6: Total surface area A_total = 784 * pi + 294 * pi + 196 * pi = 1274 * pi.Step 7: Using pi = 22 / 7, A_total = 1274 * (22 / 7) = 4004 square centimetres.

Verification / Alternative check:
As a consistency check, note that if the hemispherical cavities were not present, the total surface area would be larger, because the full base discs would be present instead of the combination of annular rings and spherical cavities.We can also recompute each area term separately and confirm that their sum is 1274 * pi before substituting the approximate value of pi.Rechecking the arithmetic for the exponents on radii and the factors of 2 for the two ends and two hemispheres ensures that no part was omitted or double counted.

Why Other Options Are Wrong:
The options 3842 sq cm, 4312 sq cm, 3296 sq cm, and 4436 sq cm do not match the carefully computed total of 4004 sq cm.They could arise from mistakes such as ignoring the annular flats, using only one hemisphere instead of two, or forgetting the curved area of the cylinder.Only 4004 sq cm is consistent with all components of the remaining surface.

Common Pitfalls:
A common mistake is to subtract hemisphere areas from the base areas instead of adding the curved cavity surfaces as additional exposed surfaces.Another pitfall is to forget that two bases and two hemispherical cavities are involved, leading to missing a factor of 2 in the calculation.Learners may also confuse volumes and surface areas, so it is important to stay focused on surface measures only in this problem.

Final Answer:
The total surface area of the remaining solid is 4004 sq cm.