91Ó°ÊÓ

1. The height of the mountain,$H=6.0km\mathrm{or}6000\mathrm{m}$
2. Thickness of a continent,$T=32kmor\mathit{}32000m$
3. Density of continental rock,${\mathrm{ÒÏ}}_{\mathrm{c}}=2.9g/c{m}^3\mathrm{or}2.9×{10}^{3}kg/m^3$
4. Density of mantle, ${\mathrm{ÒÏ}}_{\mathrm{m}}=3.3g/c{m}^3\mathrm{or}3.3×{10}^{3}kg/m^3$

The pressure at some horizontal level of compression, deep inside the earth, is the same over a large region and is equal to the pressure exerted by the gravitational force on the overlying material. Therefore, using the formula of gauge pressure, we can find the depth of the root.

Formula:

Pressure applied on a body, $p=\mathrm{ÒÏ}gh$ (i)

$$\begin{gathered} As{p}_a=p_b, \\ Fromequation\left(i\right)andthegivenvalues,weget \\ But,p_a={\mathrm{ÒÏ}}_{c}g(6.0km+32km+D)+{\mathrm{ÒÏ}}_{\mathrm{m}}(y\mathit{-}D).......\left(a\right)and \\ {p}_b={\mathrm{ÒÏ}}_{\mathrm{c}}g\mathit{\left(}\mathit{32}km\mathit{\right)}\mathit{+}{p}_{\mathit{m}}y\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\left(b\right) \\ Therefore,equatingequations\left(a\right)and\left(b\right),weget \\ {\mathrm{ÒÏ}}_{\mathrm{c}}g(6.0\mathrm{km}+32\mathrm{km}+D)+{\mathrm{ÒÏ}}_{\mathrm{m}}(y\mathit{-}D)={\mathrm{ÒÏ}}_{\mathrm{c}}g\left(32km\right)+{\mathrm{ÒÏ}}_{\mathrm{m}}y \\ {\mathrm{ÒÏ}}_{\mathrm{c}}(38.0km+D)+{\mathrm{ÒÏ}}_{\mathrm{m}}(y\mathit{-}D)={\mathrm{ÒÏ}}_{\mathrm{c}}\left(32km\right)+{\mathrm{ÒÏ}}_{\mathrm{m}}y \\ {\mathrm{ÒÏ}}_{\mathrm{c}}(6.0km)+{\mathrm{ÒÏ}}_{\mathrm{c}}D+{\mathrm{ÒÏ}}_{\mathrm{m}}y\mathit{-}{\mathrm{ÒÏ}}_{\mathrm{m}}D-{\mathrm{ÒÏ}}_{\mathrm{m}}y=0 \\ {\mathrm{ÒÏ}}_{\mathrm{c}}(6.0km)+{\mathrm{ÒÏ}}_{\mathrm{c}}D\mathit{-}{\mathrm{ÒÏ}}_{\mathrm{m}}D\mathit{=}0 \\ D\mathit{=}\frac{{\mathit{ÒÏ}}_{\mathit{c}}\mathit{(}\mathit{6}\mathit{.}\mathit{0}\mathit{k}\mathit{m}\mathit{)}}{{\mathit{ÒÏ}}_{\mathit{m}}\mathit{-}{\mathit{ÒÏ}}_{\mathit{c}}} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}D\mathit{=}\frac{\left(\mathrm{2}\mathrm{.}\mathrm{9}g/c{m}^3\right)\left(\mathrm{6}\mathrm{.}\mathrm{0}km\right)}{\mathit{3}\mathit{.}\mathit{3}\mathit{g}\mathit{/}\mathit{c}{\mathit{m}}^{\mathit{3}}\mathit{-}\mathit{2}\mathit{.}\mathit{9}\mathit{g}\mathit{/}\mathit{c}{\mathit{m}}^{\mathit{3}}} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}D=43.5km \end{gathered}$$

Therefore, the depth of the root is $43.5km$