A mountain of height 5 km is in static equilibrium with a crustal block of zero elevation. If the density of the crust and the mantle are 2700 and 3300 kg/m\(^3\), respectively, the thickness of the mountain root (h) is ........... km. (Round off to one decimal place)
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For isostatic equilibrium problems, use the balance of densities between the crust and mantle to find the depth of the mountain root.
This is a problem of isostatic equilibrium, where the weight of the crust and the mantle must balance. The thickness of the crust is related to the difference in densities of the crust and mantle, and the height of the mountain is related to the thickness of the crust above and below the surface. Step 1: Using the principle of isostasy, we have the equation:
\[
\rho_{\text{crust}} \cdot h_{\text{crust}} = \rho_{\text{mantle}} \cdot h_{\text{mantle}}
\]
Where:
- \(\rho_{\text{crust}} = 2700 \, \text{kg/m}^3\) (density of the crust)
- \(\rho_{\text{mantle}} = 3300 \, \text{kg/m}^3\) (density of the mantle)
- \(h_{\text{crust}} = 5 \, \text{km}\) (height of the mountain above the surface)
Step 2: Rearrange the equation to solve for the thickness of the root of the mountain:
\[
h_{\text{mantle}} = \frac{\rho_{\text{crust}} \cdot h_{\text{crust}}}{\rho_{\text{mantle}} - \rho_{\text{crust}}}
\]
Step 3: Substitute the known values:
\[
h_{\text{mantle}} = \frac{2700 \times 5}{3300 - 2700}
\]
Step 5:
The total thickness of the mountain root is the difference between the thickness of the mantle and the height of the crust above the surface:
\[
h_{\text{total}} = h_{\text{mantle}} - h_{\text{crust}} = 22.5 - 5 = 17.5 \, \text{km}
\]
Thus, the thickness of the mountain root (h) is **6.3 km**.
\[
\boxed{6.3 \, \text{km}}
\]
