Question

A solid is in the shape of a cone mounted on a hemisphere of same base radius.if the curved surfaces are equal then find the ratio of radius and height

Answer 1

so,
r:h=1:Root3
hope it helps..

Answer 2

Answer:

The ratio of radius to height is 1 : √3 or √3 : 3

Step-by-step explanation:

Let the radius of the hemisphere be R,

radius of the cone = r

height of the cone = h

slant height of the cone = l = $\sqrt{r^2+h^2}$

We know that curved surface area (CSA)of cone = πrl

and curved surface area(CSA) of hemisphere = 2πr²

Given that CSA of cone = CSA of hemisphere

=> πrl = 2πr²

=> rl = 2r²

=>   $\sqrt{r^2+h^2}$ = 2r

Squaring both sides,

=> r² + h² = 4r²

=> h² = 4r² - r²

=> h² = 3r²

Taking square root both sides,

=> h = √3r

=> $\frac{r}{h}$ = $\frac{1}{\sqrt{3} }$

Rationalizing the denominator

=>  $\frac{r}{h}$ = $\frac{1}{\sqrt{3} }$ *  $\frac{\sqrt{3}}{\sqrt{3} }$ =  $\frac{\sqrt{3} }{3}$

Therefore, the ratio of radius to height is 1 : √3 or √3 : 3