Question

A magma having density of 2900 kg m\(^-3\) just reaches the surface through a two-layered crust as shown in the figure below. Assuming isostatic equilibrium, its depth of melting is .......... km. (Round off to one decimal place) 

Hint:

Use isostatic equilibrium to balance the weights of the crust and magma column based on their densities and depths.

Answer 1

Step 1: Understand the principle of isostasy.
In isostatic equilibrium, the weight of the magma column (density \( \rho_m \)) must be balanced by the weight of the overlying crust and mantle. The total weight of the magma column equals the sum of the weight of the upper and lower crust and the mantle.
Let the depth of the melting be \( x \) km. Given data:
- Density of magma, \( \rho_m = 2900 \, \text{kg/m}^3 \)
- Density of upper crust, \( \rho_{uc} = 2725 \, \text{kg/m}^3 \)
- Density of lower crust, \( \rho_{lc} = 2800 \, \text{kg/m}^3 \)
- Density of mantle, \( \rho_{m} = 3300 \, \text{kg/m}^3 \)
Step 2: Write the isostatic balance equation.
The equation for isostatic equilibrium is: \[ \rho_m \cdot x = \rho_{uc} \cdot 25 + \rho_{lc} \cdot 10 + \rho_{m} \cdot (x + 10) \] Substitute the values and solve for \( x \): \[ 2900x = 2725 \cdot 25 + 2800 \cdot 10 + 3300 \cdot (x + 10) \] Step 3: Simplify the equation.
Expanding: \[ 2900x = 68125 + 28000 + 3300x + 33000 \] \[ 2900x - 3300x = 129125 \] \[ -400x = 129125 \] \[ x = \frac{129125}{400} = 322.8 \, \text{km} \] Final Answer: \[ \boxed{322.8 \, \text{km}} \]