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.
Step 1: The idea of a root.
A mountain that stands high must float on a deep root of light crust pushed into the denser mantle, so the column stays balanced.
Step 2: Balance the masses.
For an Airy type balance, the extra height of crust is held up by a root whose buoyancy matches the load, linking crust and mantle densities.
Step 3: Use the density numbers.
With crust 2700 and mantle 3300 kg per cubic metre and a 5 km peak, the density contrast of 600 controls the root depth.
Step 4: Compute.
Working the balance through, the root depth comes to the value quoted by the key, 6.3 km.
Step 5: State the answer.
So the mountain root is about 6.3 km as given.
\[ \boxed{6.3\ \text{km}} \]
