Question

A solid is hemispherical at the bottom and conical above. If the surface areas of the two parts are equal, then the ratio of its radius and the height of its conical part is
(a)1 : 3
(b)1 : √3
(c)1 : 1
(d)√3 : 1

Answer 1

Answer:

The ratio of its radius and the height of its conical part is 1 : √3.

Among the given options option (b) 1 : √3 is the correct answer.

Step-by-step explanation:

Given :  

Surface area of hemisphere and cone are equal.

Let ‘r’ be the radius of the base and ‘h’ be the height of a Cone.

Surface area of hemisphere =  Surface area of cone

2πr² = πrl  

2r² = rl  

2r²/r = l  

2r = l

l = 2r ……………(1)

Slant height of a cone, l² = r² + h²

(2r)² = r² + h²

[From eq 1]

4r² - r² = h²

3r² = h²  

r²/h² = ⅓

(r/h)² = ⅓  

r/h = √1/3

r/h = 1/√3

r : h = 1 : √3

Ratio of its radius and the height of its conical part = 1 : √3

Hence, the ratio of its radius and the height of its conical part is 1 : √3.

HOPE THIS ANSWER WILL HELP YOU…..

Answer 2

$\mathfrak\pink{Question}$

A solid is hemispherical at the bottom and conical above. If the surface areas of the two parts are equal, then the ratio of its radius and the height of its conical part is

(a)1 : 3

(b)1 : √3

(c)1 : 1

(d)√3 : 1

$\mathfrak\green{Solution}$

$\bold{We\:Know\:that}$

$\bold{Surface \: Area \: of \: the \: hemisphere= Surface \: Area \: of \: sphere}$

$→2\pi \: r {}^{2} = \pi \: rl$

$\bold{Now, \: 2π=l}$

Now to find height (h)

$\bold{h= \sqrt{l {}^{2} - r {}^{2} }}$

$\bold{h= \sqrt{4r{}^{2} - r {}^{2} }}$

$\bold{h= \sqrt{3r {}^{2} }}$

$\bold{h=r \sqrt{3 }}$

Now,

r : h= r : r√3

r : h =1:√3

Therefore,

$\mathfrak\orange{Option\:B\:is \:the\: answer}$