When studying the formation of mountain ranges geologists es

When studying the formation of mountain ranges, geologists estimate the amount of work required to lift a mountain from sea level. Consider a mountain that is essentially in the shape of a right circular cone, with base radius R and height H. At some interior point located at (x, y, z), let the mass density of the material be delta (x, y, z), and let the height at the interior point be h(x, y, z). Determine the definite triple integral that represents the total work done in forming the mountain. Your answer should be written in terms of R, H, delta (x, y, z), and h(x, y, z). Assume that Mount Fuji in Japan is in the shape of a right circular cone with a base radius of 18.9 km, height of 3.8 km, and density of 2.7 g/cm^3. How much work was done in forming Mount Fuji if the land was initially at sea level? Express your answer in kJ.

Solution

You need to integrate in cylindrical coordinates. If r is the radial coordinate (horizontal), z is the axial coordinate (vertical), and is the angular coordinate, then you need to do a triple integral.

The work function is m*g*h, where m = mass, g = gravity, h = height. The mass of a differential volume of rock \"dV\" is *dV, where the = density, and dV = r*dr*dz*d, which is the differential volume in cylindrical coordinates. So, the work done raising a differential volume of rock a distance z above sea level is *g*z*r*dr*dz*d.

We are integrating all the way around in a circle, so its limits of integration are 0 to 2*.

We are integrating z from 0 to H, where H is the height of the mountain.

If it was a cylinder, we would be integrating r from 0 to H as well, but we need to take into account that the shape is a cone, not a cylinder. So, the distance along which we integrate r will change with z. Instead, we integrate r from 0 to (H - z). That means when we\'re at bottom of the mountain, we integrate r from 0 to H, but when we\'re at the top, we don\'t even integrate at all, since the mountain has zero width at the very tip.

It boils down to this triple integral, given in the form
Int[ function, variable, lower limit, upper limit ]

Total Work = Int[ Int[ Int[ *g*z*r, r, 0, H-z ], z, 0, H ], , 0, 2* ]

Performing all of this, you should end up with this solution:

Total Work = (1/12)***g*H^4

2)w=1278678.39J