Q) A solid is in the shape of a cone mounted on a hemisphere of same base radius If the - Maths - Surface Areas and Volumes

Question

Q) A solid is in the shape of a cone mounted on a hemisphere of
same base radius If the curved surface areas of teh hemispherical
part and the conical part are equal,then find the ratio of the
radius and the height of the conicall part?
Q) A sphere of diameter 6cm is dropped into cylindrical
vessel,partly filled with water ,whose diameter is 12cm If the
sphere is completely submerged in water , by how much time will the
surface of water be raised in the cylindrical vessel?

Santosh Kumar · Accepted Answer

Let the common radius be r and height of the conical part be h.

Curved surface area of the hemispherical part = 2πr²

Curved surface area of the conical part = πrl

where, l = slant height

$l = \sqrt{h^{2} + r^{2}}$

∴ curved surface area of the conical part $= π r \sqrt{h^{2} + r^{2}}$

Now,

As curved surface areas are equal.

$∴ 2 π r^{2} = π r \sqrt{h^{2} + r^{2}} \Rightarrow 2 r = \sqrt{h^{2} + r^{2}} \Rightarrow {(2 r)}^{2} = h^{2} + r^{2} \Rightarrow 4 r^{2} = h^{2} + r^{2} \Rightarrow 4 r^{2} - r^{2} = h^{2} \Rightarrow 3 r^{2} = h^{2} \Rightarrow \frac{r^{2}}{h^{2}} = \frac{1}{3} \Rightarrow \frac{r}{h} = \frac{1}{\sqrt{3}} \Rightarrow r : h = 1 : \sqrt{3}$

20