Of all right circular cones with slant height 6√3 cm, the one with th…

The correct answer is Option (3) → \(144\pi\) cm³

Let the height be h and the base radius r. The slant height is \(6\sqrt{3}\), so

    \( r^{2} = (6\sqrt{3})^{2} - h^{2} = 108 - h^{2} \)

The volume as a function of h is

    \( \displaystyle V = \frac{1}{3}\pi r^{2}h = \frac{\pi}{3}\left(108h - h^{3}\right) \)

Setting the derivative equal to zero:

    \( \displaystyle \frac{dV}{dh} = \frac{\pi}{3}\left(108 - 3h^{2}\right) = 0 \), so h = 6

Since \(\frac{d^{2}V}{dh^{2}} = -2\pi h \lt 0\), this is a maximum. Then \(r^{2} = 108 - 36 = 72\), and

    \( \displaystyle V = \frac{1}{3}\pi \times 72 \times 6 = 144\pi \) cm³