Conical hemisphere?

Geometry Level 3

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

1:3

2:\sqrt{3}

1:\sqrt{3}

1:4

1 solution

Rishik Jain
Feb 16, 2016

Radius of the cone=Radius of the hemisphere

Surface area of the cone=Surface area of the hemisphere

$\begin{aligned} \pi \cdot r \cdot l=2 \pi \cdot r^2 \\ l=2r \\ l=\sqrt{r^2 +h^2} \\ (2r)^2=r^2+h^2 \\ \left(\frac{r}{h}\right)^2=\frac{1}{3} \\ \dfrac{r}{h}=\large \boxed {\dfrac{1}{\sqrt{3}}} \end{aligned}$

rishik, how do you know that their radii are equal? ?

Yash Mehan - 5 years, 3 months ago

If you create a figure, you will come to know that the cone is placed exactly on top of the hemisphere. That means they both have a common base which tells us they have equal radii.

Rishik Jain - 5 years, 3 months ago

Yeah. . Just forgot about all these questions

Yash Mehan - 5 years, 3 months ago

Conical hemisphere?

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