3 q9.s17.pdf 4. Triple Integral Problem When studying the formation of mountain

Question

3 q9.s17.pdf 4. Triple Integral Problem 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. Suppose that the weight density of the material in the vicinity of a point P is g(P) and the height is h(P). (a) (5 points) Find a definite integral that represents the total work done in forming the mountain. (h) (h points) A Monnt Fuji in lapai is in the shape a right circu ar cone with radiIN 02,000 ft. lleight 12.100 ft and density a 200 ibi How work was doue in formuiug Mount Fuji if the

Explanation / Answer

1. 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
(where everything is in SI units)

2. given r= 62000 ft

H =12400 ft rho =200 lb/ft^3

therefore work done = 4.010794153 * 10^19 ft-lb

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