The radius of a cone is √2 times the height of the cone. A cube of m

1.	The radius of a cone is √2 times the height of the cone. A cube of maximum possible volume is cut from the same cone. What is the ratio of volume of the cone to the volume of the cube?(a) 3.18 π(b) 2.25 π (c) 2.35 π(d) None of these
Answer» Answer : (b) = 2.25 π Let each side of the cube be a cm. Then DE = diagonal of square face = √2a ⇒ DG = \(\frac{\sqrt{2}a}{2}\) = \(\frac{a}{\sqrt{2}}\) cm. Let the radius of the cone, i.e., AF = FB = r cm. Height FC = h cm. Given, r = h√2 In similar triangles, AFC and DGC \(\frac{AF}{FC}=\frac{DG}{GC}\) ⇒ \(\frac{r}{h} = \frac{\frac{a}{\sqrt{2}}}{(h-a)}\) ⇒ \(\frac{a/\sqrt{2}}{h-a} = \sqrt{2} \) ⇒ a = 2(h-a) ⇒ h = \(\frac{3a}{2}\) , r = \(\frac{3a}{2}\) \(\times \,\sqrt{2}\) ∴ Volume of cone : Volume of cube = \(\frac{\frac{1}{3}\pi \times \big(\frac{3a\sqrt{2}}{2}\big)^2 \times \frac{3a}{2}}{a^3}\) = \(\frac{9}{4} a^3\pi :a^3 = \frac{9}{4}\pi\) = 2.25 π

The radius of a cone is √2 times the height of the cone. A cube of maximum possible volume is cut from the same cone. What is the ratio of volume of the cone to the volume of the cube?(a) 3.18 π(b) 2.25 π (c) 2.35 π(d) None of these

Answer»

Answer : (b) = 2.25 π

Let each side of the cube be a cm.

Then DE = diagonal of square face = √2a

⇒ DG = \(\frac{\sqrt{2}a}{2}\) = \(\frac{a}{\sqrt{2}}\) cm.

Let the radius of the cone, i.e., AF = FB = r cm.

Height FC = h cm.

Given, r = h√2

In similar triangles, AFC and DGC

\(\frac{AF}{FC}=\frac{DG}{GC}\) ⇒ \(\frac{r}{h} = \frac{\frac{a}{\sqrt{2}}}{(h-a)}\)

⇒ \(\frac{a/\sqrt{2}}{h-a} = \sqrt{2} \)

⇒ a = 2(h-a) ⇒ h = \(\frac{3a}{2}\) , r = \(\frac{3a}{2}\) \(\times \,\sqrt{2}\)

∴ Volume of cone : Volume of cube = \(\frac{\frac{1}{3}\pi \times \big(\frac{3a\sqrt{2}}{2}\big)^2 \times \frac{3a}{2}}{a^3}\)

= \(\frac{9}{4} a^3\pi :a^3 = \frac{9}{4}\pi\)

= 2.25 π