Question

The ratio of the radii of two right circular cones of the same height is 1:2. Find the ratio of their :
1. Volumes
2. Curved surface areas when the height of each cone is 3 times the smaller radius. ​

Answer 1

Step-by-step explanation:

${\Large{\mathbf{\red{Question :-}}}}$

The ratio of the radii of two right circular cones of the same height is 1:2. Find the ratio of their :

1. Volumes

2. Curved surface areas when the height of each cone is 3 times the smaller radius.

${\Large{\mathbf{\red{Solution :-}}}}$

Let the radii be r and R respectively, and let h be the height of each cone

According to question,

${\large{\red{✠ \: \: \: \: \: }}}{\bold{\sf{\frac{r}{R} \: = \: \frac{1}{2}}}}$

${\bold{\sf{\orange{ \: \: \: \: \: R \: \: = \: \: 2r}}}}$

1. Let the volumes be v and V respectively

Therefore,

${\bold{\sf{: \: ⟹ \: \: \: \: \: \: v \: \: = \: \: \frac{1}{3}πr²h}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: \: V \: = \: \frac{1}{3}πR²h}}}$

We know that R = 2r,

${\bold{\sf{: \: ⟹ \: \: \: \: \: \: V \: = \: \frac{1}{3}π(2r)²h}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: V \: = \: 4\Big(\frac{1}{3}πr²h\Big) \: = \: 4v}}}$

${\bold{\sf{: \: ⟹ \: \: \: \: \: \: \frac{v}{V} \: = \: \frac{v}{4v} \: = \: \frac{1}{4}}}}$

$: \: ⟹ \: \: \: \: \: \: {\bold{\mathfrak{\fbox{\purple{v : V \: \: = \: \: 1 : 4}}}}}$

2. Let the curved surface areas be s and S respectively

Here,

h= 3r and R = 2r

✯Let's find the curved surface area

For s,

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \: s \: = \: πrl }}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \: s \: = \: πr\sqrt{h² \: + \: r²} }}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \: s \: = \: πr\sqrt{(3r)² \: + \: r²} }}}$

$: \: ⟹ \: \: \: \: \: \: {\bold{\mathrm{\orange{ \: \: \: \: \: \: s \: = \: π \: \times \: \sqrt{10r²}}}}}$

For S,

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \: S \: = \: πRL }}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \: S \: = \: π × 2r\sqrt{h² + R²}}}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \: S \: = \: π \: × \: 2r\sqrt{(3r)² \: + \: (2r)²} }}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \: \: \: S \: = \: 2πr\sqrt{13r} }}}$

$: \: ⟹ \: \: \: \: \: \: {\bold{\mathrm{\orange{ \: \: S \: = \: 2π\sqrt{13r²} }}}}$

✯Now we have both the curved surface areas, so let's calculate their ratios

${\bold{\mathbf{: \: ⟹ \: \: \: \: \: \: \frac{s}{S} \: = \: \frac{π\sqrt{10r²}}{2π\sqrt{13r²}}}}}$

${\bold{\mathbf{: \: ⟹ \: \: \: \: \: \frac{s}{S} \: = \: \frac{\sqrt{10}}{2\sqrt{13}}}}}$

✯The ratio of their curved surface area is :

$: \: ⟹ \: \: \: \: \: \: {\bold{\boxed{\mathfrak{\pink{ s : S \: \: = \: \: \sqrt{10} \: : \: 2\sqrt{13}}}}}} \\ </p><p>$