Using Airy's Model of isostatic equilibrium, solve for....

[Nicki]

Homework Statement

Using Airy's Model of isostatic equilibrium, solve for the average anti-root height due to the ocean and the average density of the upper mantle.

given info: ocean depth avg = 4km
ocean crust is 10 km thick avg
continental crust if 40 km thick avg
all crust is the same density (2.5g /cm³), density of water is (1g/cm³)
[[[[We can't use the value given for the mantle density in the image]]]]

Homework Equations

The Attempt at a Solution

I know the equation to find the avg height of the anti-root, but you need the density of the mantle to solve for it..? This isn't given to us

The next question is to solve for the density of the mantle, and from the equation manipulation I've done, you need the height of the anti root to solve for this?
I feel like I'm missing something super obvious.

Mantle density (gravity is left out because it appears in all terms)
ρc = crust density, ρw = density water ρm = mantle density

ρwh2 + ρc(C-h2-d2) + ρm(b2+b1) = ρcC + ρmb1
ρwh2 + ρmb2 = ρch2 + ρcb2
ρm = (ρcC + ρcb2 - ρwh2)/ b2

you need b2 (the anti-root height) to solve this, and you need the ρm to solve for anti root height? I feel like I'm missing something very obvious?

 

[nate808]

You are correct that you need the density of the mantle to solve for the average anti-root height. However, you can still solve for the density of the mantle using the given information and Airy's Model of isostatic equilibrium. Here is a step-by-step guide to solving this problem:

1. Start by writing down the equation for Airy's Model of isostatic equilibrium:

ρwh2 + ρc(C-h2-d2) + ρm(b2+b1) = ρcC + ρmb1

2. Plug in the given values for ocean depth (h), ocean crust thickness (d), continental crust thickness (C), and crust density (ρc). This will give you an equation with two unknowns: the mantle density (ρm) and the anti-root height (b2).

3. Rearrange the equation to solve for b2:

b2 = (ρcC + ρcb2 - ρwh2)/ρm

4. Now, we can substitute the given values for ocean depth (h) and ocean crust thickness (d) to get:

b2 = (ρc(40km) + ρcb2 - ρw(4km))/ρm

5. Next, we can substitute the given value for crust density (ρc = 2.5 g/cm3):

b2 = (2.5g/cm3(40km) + 2.5g/cm3b2 - 1g/cm3(4km))/ρm

6. Simplify the equation to get:

b2 = (100g/cm2 + 2.5g/cm3b2 - 4g/cm2)/ρm

7. Now, we can plug in the given value for the average ocean crust thickness (10 km) to get:

b2 = (100g/cm2 + 2.5g/cm3(10km) - 4g/cm2)/ρm

8. Simplify the equation to get:

b2 = (50g/cm2)/ρm

9. Finally, we can solve for the density of the mantle (ρm):

ρm = (50g/cm2)/b2

10. Plug in the given value for the average anti-root height (b2) to get your final answer for the mantle density.

I hope this helps you understand how to solve for