A rectangle of perimeter 36 cm is revolved about one of its sides to…

The correct answer is Option (1) → \(864\pi\) cm³

Let the rectangle be revolved about a side of length y. The other side x becomes the radius and y the height, with

    \( 2(x + y) = 36,\ y = 18 - x \)

The volume is

    \( V = \pi x^{2}(18 - x) = \pi(18x^{2} - x^{3}) \)

Setting the derivative equal to zero:

    \( V' = \pi(36x - 3x^{2}) = 3\pi x(12 - x) = 0,\ x = 12 \)

Since \(V'' = \pi(36 - 6x) = -36\pi \lt 0\) at x = 12, this is the maximum. Then y = 6 and

    \( V = \pi \times 144 \times 6 = 864\pi \) cm³

Why the other options are incorrect

• Option (2) \(729\pi\) cm³ — Incorrect. This is the square x = y = 9: \(\pi \times 81 \times 9\); the square does not give the largest cylinder here.

• Option (3) \(432\pi\) cm³ — Incorrect. This is radius 6 and height 12: \(\pi \times 36 \times 12\), the best rectangle revolved about the wrong side.

• Option (4) \(1296\pi\) cm³ — Incorrect. The largest value of \(\pi x^{2}(18 - x)\) is \(864\pi\), at x = 12.