Question

a cone surmounted on a hemisphere of same radii. if the surface area of a cone and hemisphere are equal find the ratio of the radius and height of the conical part

Answer 1

Answer:

l=2r. ......may be...

Answer 2

The ratio of  the radius and height i.e. r:h is  1:√3

Step-by-step explanation:

Given that radius of cone is = radius of hemisphere and surface area of cone = surface area of hemisphere

To find the ratio of  radius and  the height of the conical part

Consider

Radius of cone = radius of hemisphere= r

Height of cone = h

Slant height of cone = l

Cone is surmounted on sphere

Therefore we know that

Curved surface area of cone = Curved surface area of hemisphere

πrl= 2πr²

l=2r

we know that

$l=\sqrt{h^2+r^2} \\ \Rightarrow \sqrt{h^2+r^2}= 2r\\ \text { Squaring both side we get }\\ \Rightarrow h^2+r^2=(2r)^2\\ \Rightarrow h^2+r^2=4r^2\\ \Rightarrow h^2=4r^2-r^2\\\Rightarrow h^2=3r^2 \\\\\Rightarrow (\dfrac{h}{r} )^2 =3 \\\\\Rightarrow \dfrac{r}{h} =\dfrac{1 }{\sqrt{3} }$

Therefor the ratio r:h= 1:√3

#Learn more

The ratio of the volume of a right circular cylinder and a right circular cone of the same base and height will be:

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