Name: Area of Quadrilateral Using Heron's Formula
Uploaded: 2019-07-19
Duration: 10 min 24 s
Description: Did you know that the area of a quadrilateral can be calculated by dividing the quadrilateral into two triangles and using Heron's formula? Let's watch this...

Question

A solid sphere is melted and recast into a cone whose (base) radius is the same as that of the sphere. Show that the slant height of the cone bears a ratio

\sqrt{17} : 4

to its height.

Accepted Answer

Volume of sphere $= \frac{4}{3} {πr}^{3}$

Volume of cone $= \frac{1}{3} {πr}^{2} h$

The sphere is melted and recast into the cone. So the volume of the sphere is equal to the volume of the cone.

The volume of sphere $=$ Volume of cone

$\Rightarrow \frac{4}{3} {πr}^{3} = \frac{1}{3} {πr}^{2} h$

$\Rightarrow 4 r = h$

$\Rightarrow r = \frac{h}{4}$

We know that slant height $(l) = \sqrt{h^{2} + r^{2}}$

$\Rightarrow l = \sqrt{h^{2} + {(\frac{h}{4})}^{2}}$

$\Rightarrow l = \sqrt{h^{2} + \frac{h^{2}}{16}}$

$\Rightarrow l = \sqrt{\frac{16 h^{2} + h^{2}}{16}}$

$\Rightarrow l = \sqrt{\frac{17 h^{2}}{16}} = \frac{\sqrt{17} h}{4}$

The ratio of slant height to height $=$ $\frac{\sqrt{17} h}{4} : h = \frac{\sqrt{17}}{4}$

A solid sphere is melted and recast into a cone whose (base) radius is the same as that of the sphere. Show that the slant height of the cone bears a ratio $\sqrt{17} : 4$ to its height.

Important Questions on Mensuration

Learn and Prepare for any exam you want !

A solid sphere is melted and recast into a cone whose (base) radius is the same as that of the sphere. Show that the slant height of the cone bears a ratio 17:4 to its height.

Important Questions on Mensuration

Learn and Prepare for any exam you want !

A solid sphere is melted and recast into a cone whose (base) radius is the same as that of the sphere. Show that the slant height of the cone bears a ratio $\sqrt{17} : 4$ to its height.