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Seattle’s factory is the company’s newest. It was constructed with sustainable building techniques and materials, so its cost structure has been affected in several ways. The Seattle plant uses a mix of solar power (not very reliable with so many clouds), thermal power, and wind. It has an array of batteries, so at the beginning of each production day, there is “free” stored energy that can produce up to 8,000 units. Of course, energy is just one factor of the cost of production, but it is significant.

The cost function for the Seattle factory is given by the following piecewise function. That is, C(x)is the cost function, where x is the number of units produced. The plant’s maximum capacity is 42,000 units per day.

$$C(x)=\left \{ {{0.35 if x \leq 8,000} \atop {0.75 if 8,000\ \textless \ x\leq 20,000}} \right \atop0.83-(\frac{x}{200,000} ) if 20,000\ \textless \ x\leq 42,000$$

A) Sketch a graph to model Seattle’s cost structure over the domain [0, 42000]. Be sure to label the axes and any endpoints where the graph breaks.

B) Describe the function over each part of its domain. State whether it is constant, increasing, or decreasing, and state the slope over each part.

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Natural Capitalism as an idea and thesis for a book emerged in 1994, the year after the

publication of The Ecology of Commerce. After meeting with and speaking to different

publication, it became clear to Hawken that industry and government needed an overall

biological and social framework within which the transformation of commerce could be

accomplished and practiced. To that end, articles and papers were written that became the

basis of a book about natural capitalism. A key element of this theory was the idea that

the economy was shifting from an emphasis on human productivity to a radical increase

in resource productivity. This shift would provide more meaningful family-wage jobs, a

better worldwide standard of living to those in need, and a dramatic reduction of

humankind's impact upon the environment. So while the context for Natural Capitalism

existed in a theoretical framework, the exposition did not.

Contemporaneously, Amory and Hunter Lovins were coming to the same conclusion: that

a shared framework was needed that could harness the talent of business to solve the

world's deepest environmental and social problems. Both were writing Factor Four:

Doubling Wealth, Halving Resource Use for publication in Germany in 1995. The senior

author of Factor Four, Ernst von Weizsäcker, among Europe's top innovators in

environmental policy, had teamed up with the Lovinses to pool the experience of their

respective nonprofit research centers—Wuppertal Institute in Germany and Rocky

Mountain Institute (RMI) in Colorado. The three authors had assembled fifty case studies

of at least quadrupled resource productivity to detail how, across whole economies,

people could live twice as well but use half as much material and energy. Factor Four

showed that such striking gains in resource efficiency could be profitable, and that

obstacles to their implementation could be hurdled by combining innovations in business

practice and in public policy.

Both Factor Four and The Ecology of Commerce urged the private sector to move to the

vanguard of environmental solutions. Factor Four described a creative policy framework

that could foster fair and open competition in achieving that success. The Ecology of

Commerce suggested techniques that when combined with business's unique strengths

could enable it to meet this challenge successfully.

Hunter Lovins sent a draft of Factor Four to Paul Hawken in early 1995. He saw that it

was the exposition that natural capitalism needed if it were to make its theoretical claims

credible and demonstrable. The ideas not only meshed, they were absolutely

complementary. We agreed to work together toward one book, under the title of Natural

Capitalism, that would contain both theory and practice. After the work began, we

discovered it wasn't that simple. Factor Four was anecdotal, Europe-oriented (by 1997 it

had also been published in England after being a German bestseller for nearly two years),

and written more for policy and environmental activists than for business practitioners. It

needed not adaptation but complete rewriting. Further, the examples offered concentrated

mainly on efficiency and did not take fully into account the need for the restoration of

natural capital nor for several other important elements of natural capitalism that go far
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