To be clear: this is not a question about whether a 30km mountain on an earth like planet could exist. It is a question about how it would look.
I found this fine math laden site which shows the maximum height for a mountain made of granite.
The density of granite is ρ = 3 g/cm3 (actually, the densities of most
liquids and solids are close to 1. Lead is only about 11 g/cm3 and
gold is 19.3 g/cm3). The total weight of the mountain is just the
volume times density times g so:
Weight W≈ ρgr2h
To see when the rock
will start to break, we’ll compare the stress of the weight of the
mountain to the compressive strength of granite. (Most mountains
aren’t made out of granite, but it should give us a good upper limit
on mountain heights). The weight of the mountain is spread out over
an area of roughly (ignoring constants such as π): A ≈ r2 so the
stress σ the mountain exerts on the ground underneath it is:
σ ≈ W/A ≈ (ρ g r2h)/r2 ≈ ρgh
The compressive strength of a material is the maximum compressive >stress a material can withstand before it starts to deform.
For granite the compressive strength is σC = 200
megaPascals = 2 × 108 N/m2 so the rock beneath the mountain will start
to compress when: σ = σC or ρghmax = σC. Rearrange this equation to
solve for hmax yields: hmax ≈ σC/(ρg) The max height for a mountain
works out to be:
hmax ≈ 2×108 N/m2/(3×103 kg/m3 ˙ 10 m/s2 )≈ 104 m = 10 km
So a granite mountain can only be 10 km. A mountain on earth which was 30 km must be made of material that is less dense, or which has a higher compressive strength.
Less dense is a nonstarter because granite is not that dense at 3, and less dense materials have markedly less compressive strength.
More compressive strength is a tall order because granite is the best among stones at 200. So not stone.
Sapphire is more dense than granite at 3.98 (we will use 4) instead of 3 g/cc. But the compressive strength is 2 GPa - that is 2000 MPa or an order of magnitude greater than granite.
Plugging in these new values
hmax ≈ 20×108 N/m2/(4×103 kg/m3 ˙ 10 m/s2 )≈ 754 m = maximum of 75 km
So 30 km is fine. This mountain would not necessarily be a single crystal of sapphire. But that would work. So here it how it would look.
Except for those birds. Nothing is flying up there. At all.
Also I object to that curly mountain at the side. But the principle stands.
I was thinking that maybe it is unrealistic to have a giant sapphire crystal. Where would it come from? A mountain of diamond seems so trite, but really it would be better in many respects. Lets get it over with.
So diamond: density of 3.5 and compressive strength of 60 GPA; maximum mountain height is 196 km
These diamonds would have been formed in the atmosphere of a ancient gas giant and then incorporated in the crust of this Earthlike world. I envision this huge, partly fused mountain of diamond extending farther down below the surface than it does above. Despite the huge mass balanced on one point, it does not sink further down because the bottom of the mountain is floating in denser, partly metallic molten materials. The diamond mountain is essentially an iceberg in the crust.
This also gives insight into how this diamond iceberg mountain would look. There would be no plants on it and no ice or snow. Diamond is one of the best thermal conductors there is. With its big bottom side down into the mantle, the entire thing would be very hot.
my own assembly. mount hood with uncut diamond