Chapter 15.7, Problem 31E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook 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) Find a definite integral that represents the total work done in forming the mountain.

(b) Assume that Mount Fuji in Japan is in the shape of a right circular cone with radius 62,000 ft, height 12,400 ft, and density a constant 200 lb/ft^{3}. How much work was done in forming Mount Fuji if the land was initially at sea level?

(a)

To determine

**To find:** The definite integral that represents the total work done in forming the mountains.

Explanation

**Given:**

The weight density of the mountain is

The height of the mountain is

**Definition used:**

“Total work done is the product of the mass of the given particle raised to the height and the weight density of that particle.”

**Calculation:**

By the definition stated above, the total work done is the product of the mass of the mountain raised to the height

Mass of the mountain raised to the height is nothing but the volume of the mountain at the point *P* denoted by *dV*

(b)

To determine

**To find:** The work done in forming the Mount Fuji.

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