A decorative column is made in the shape of a regular hexagonal prism. The side length of its base is 10 cm. The column is built from multiple sections, and their heights form an arithmetic progression: the first section is 5 cm tall, and each subsequent section is 2 cm taller than the previous one. If the column has 4 sections in total, what is the total volume of the column?

Correct — B. Volume of a prism = base area × height, and all four sections stand on the same hexagonal base.

Base area = (3√3⁄2) × side² = (3√3⁄2) × 100 = 150√3 cm²

Section heights, adding 2 cm each time: 5, 7, 9, 11
Total height = 5 + 7 + 9 + 11 = 32 cm

Volume = 150√3 × 32 = 4800√3 cm³ → option (b)

Why the others are wrong

• (a)
  4500√3 cm³ — 4500√3 ÷ 150√3 = 30 cm of total height. The four sections add to 5 + 7 + 9 + 11 = 32 cm, so this falls 2 cm short.
• (c)
  5100√3 cm³ — 5100√3 ÷ 150√3 = 34 cm of total height, 2 cm more than the 32 cm the four sections add to.
• (d)
  5400√3 cm³ — 5400√3 ÷ 150√3 = 36 cm of total height, 4 cm more than the 32 cm the four sections add to.

Two formulas meet here. A regular hexagon splits into six equilateral triangles, so its area is 6 × (√3⁄4)s² = (3√3⁄2)s².

The heights form an arithmetic progression with a = 5, d = 2, n = 4. Its sum is n⁄2 × (2a + (n − 1)d) = 2 × (10 + 6) = 32.

Prisms stacked on the same base make one prism whose height is the sum of the heights.

Key facts

• Area of a regular hexagon of side s = (3√3⁄2)s².
• Volume of a right prism = base area × height.
• Sum of the first n terms of an AP = n⁄2 × (2a + (n − 1)d).

Common traps

• Using (√3⁄4)s², one triangle's area, for the whole hexagon, which gives a sixth of the volume.
• Taking the fourth section's height (11 cm) as the height of the whole column.

No directly related past PYQ was found.