Sphere Problem, Solid Geometry

[Physicsrapper]

10 000 cubic km of water can cover all the continents with a layer 7 cm deep.

Find the depth of the water layer if 10 000 cubic km of water is spread over a sphere with radius R = 6370 km.

I already tried everything, but still didn't get the right answer.
The solution must be d = 2 cm

I know, that it's easy when you divide the volume by the surface area of the sphere.

Volume of the sphere: 4/3*π*r^3 ---> is 10 000 km^3 the volume?
Surface of the sphere: 4*π*r^2 ---> it's 4/3*π*6370^3 = 509.9 * 10^6 , isn't it?

So I wanted to devide 10 000 km^3 by 509.9 * 10^6, but it doesn't funcion.

What did I wrong?

 

[HallsofIvy]

10 000 cubic km of water can cover all the continents with a layer 7 cm deep.

Find the depth of the water layer if 10 000 cubic km of water is spread over a sphere with radius R = 6370 km.

I already tried everything, but still didn't get the right answer.
The solution must be d = 2 cm

I know, that it's easy when you divide the volume by the surface area of the sphere.

No, that's wrong. The would give you the height of a rectangular solid above that area, not a portion of a sphere.

Volume of the sphere: 4/3*π*r^3 ---> is 10 000 km^3 the volume?
Surface of the sphere: 4*π*r^2 ---> it's 4/3*π*6370^3 = 509.9 * 10^6 , isn't it?

So I wanted to devide 10 000 km^3 by 509.9 * 10^6, but it doesn't funcion.

What did I wrong?

You are not taking into account that the earth+ water is still a sphere.

If we call the height of the water above the surface of a sphere "r", then the water plus earth is a sphere of volume \(\displaystyle (4/3)\pi (6370+ r)^3\). Since the volume of the earth alone is \(\displaystyle (4/3)\pi (6370)^3\) the volume of water alone is \(\displaystyle (4/3)\pi (6370+ r)^3- (4/3)\pi (6370)^3= 10000\). Solve that for r.

 

[Deleted member 4993]

10 000 cubic km of water can cover all the continents with a layer 7 cm deep.

Find the depth of the water layer if 10 000 cubic km of water is spread over a sphere with radius R = 6370 km.

I already tried everything, but still didn't get the right answer.
The solution must be d = 2 cm

I know, that it's easy when you divide the volume by the surface area of the sphere.

Volume of the sphere: 4/3*π*r^3 ---> is 10 000 km^3 the volume?
Surface of the sphere: 4*π*r^2 ---> it's 4/3*π*6370^3 = 509.9 * 10^6 , isn't it?

So I wanted to devide 10 000 km^3 by 509.9 * 10^6, but it doesn't funcion.

What did I wrong?

Assume that the new sphere (after being covered with water) has a radius of = 6370+d km (depth of water = d km)

Volume of water= 10000 = 4/3 * π [(6370+d)³ - 6370³] .... now solve for 'd'

The above equation would involve a cubic function - thus you would need to calculate the roots of a cubic function. That may need "approximation" through numerical methods.