Question

A given right circular cone has a volume p, and the largest right circular cylinder that can be inscribed in the cone has a volume q. Then p : q is –

• A. 9 : 4
• B. 8 : 3
• C. 7 : 2
• D. None of these

Answer 1

Answer: (A)

9 : 4

Answer 2

Let H be the height of the cone and α be its semi vertical angle. Suppose that x is the radius of the inscribed cylinder and h be its height h = QL = OL – OQ = H – x cot α

V = volume of the cylinder

= πx² (H – x cot α)

Also p = $\frac{1}{3}$ π (H tan α)² H (i)

$\frac{dV}{dx}$ = π (2Hx – 3x² cot α)

So $\frac{dV}{dx}$ = 0 ⇔ x = 0, x = $\frac{2}{3}$ H tan α

⇒ $\left. \ \frac{d^{2}v}{dx^{2}} \right|_{x = \frac{2}{3}H\tan\alpha} = –2\pi H < 0$

∴ v is maximum when x = $\frac{2}{3}$ H tan α 

and q = V_max = π $\frac{4}{9}$H² tan² α $\frac{1}{3}$H

= $\frac{4}{9}$ p. [using (i)]

Hence p : q = 9 : 4.