A right prism has a regular hexagonal base of side 4 centimetres and height 8 centimetres. What is the total surface area of the prism, in square centimetres (cm^2)?

Introduction / Context:
This question asks for the total surface area of a right prism whose base is a regular hexagon. We are given the side length of the hexagonal base and the height of the prism. Total surface area includes the areas of the two congruent hexagonal bases and the lateral rectangular faces that connect them. The problem tests familiarity with area formulas for regular hexagons and for rectangles, as well as the structure of a prism.

Given Data / Assumptions:

• The base is a regular hexagon with side length s = 4 cm.
• The prism height (distance between the two bases) is h = 8 cm.
• The prism is right, so lateral faces are rectangles perpendicular to the bases.
• We must find total surface area in cm^2.
• Use the standard area formula for a regular hexagon.

Concept / Approach:
A regular hexagon of side s can be divided into six equilateral triangles each of side s. The area of an equilateral triangle with side s is (√3 / 4) * s^2. Therefore the area of a regular hexagon is:
Area_hexagon = 6 * (√3 / 4) * s^2 = (3√3 / 2) * s^2.
For a prism, total surface area = 2 * (area of base) + (perimeter of base) * height. We use the hexagon area formula and the perimeter of a regular hexagon (6s) to compute the total.

Step-by-Step Solution:
Step 1: Compute the area of the regular hexagonal base using s = 4 cm. Step 2: Area_hexagon = (3√3 / 2) * s^2 = (3√3 / 2) * 4^2 = (3√3 / 2) * 16 = 24√3 cm^2. Step 3: The prism has two such bases, so total area of both bases = 2 * 24√3 = 48√3 cm^2. Step 4: Perimeter of the regular hexagon = 6 * s = 6 * 4 = 24 cm. Step 5: Lateral surface area = perimeter * height = 24 * 8 = 192 cm^2. Step 6: Total surface area = area of both bases + lateral surface area = 48√3 + 192 cm^2. Step 7: Factor the expression to match the options: 48√3 + 192 = 48(√3 + 4) = 48(4 + √3) cm^2.

Verification / Alternative check:
We can double check the base area by computing the area of one equilateral triangle and multiplying by 6. For s = 4, area of one equilateral triangle is (√3 / 4) * 16 = 4√3 cm^2. For six such triangles, the hexagon area = 6 * 4√3 = 24√3 cm^2, matching our earlier computation. Adding twice this area to the correctly computed lateral area 192 again gives 48√3 + 192, confirming the total surface area.

Why Other Options Are Wrong:
Expressions like 54(3 + √3), 36(3 + √3), 24(4 + √3) and 60(2 + √3) either have incorrect coefficients for the base area, the lateral area, or both. They do not simplify to 192 + 48√3. Only 48(4 + √3) exactly expands to 192 + 48√3, which is consistent with the correct geometric calculations.

Common Pitfalls:
Students sometimes use the wrong base area formula, for example treating the hexagon as if it were made of rectangles instead of equilateral triangles. Others might forget to include both bases in the total surface area or miscalculate the perimeter, using 4s instead of 6s. Another common error is to confuse lateral area with total surface area. Carefully applying the correct formula for a regular hexagon and adding base and lateral areas separately prevents these mistakes.

Final Answer:
The total surface area of the prism is 48(4 + √3) cm^2.