A Company Manufactures a Product Using X, Y and Z $Q = 60 x ^ { \frac { 1 } { 3 } } y ^ { \frac { 1 } { 4 } } z ^ { \frac { 2 } { 5 } }$

question 77

Essay

A company manufactures a product using x, y and z units of three different raw materials.The quantity produced is given by The production is described by the function $Q = 60 x ^ { \frac { 1 } { 3 } } y ^ { \frac { 1 } { 4 } } z ^ { \frac { 2 } { 5 } }$ .Suppose the cost of the materials per unit is $20, $15 and $24 respectively.
(a)Find the cheapest way to produce 6300 units of the product.
(b)Find the value of $\lambda$ in $\nabla f = \lambda \nabla Q$ and interpret this value.

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Definitions:

Factory Worker

An individual who works in a manufacturing facility, responsible for operating machinery, assembling products, or supervising production processes.

Factory System

The factory system is an industrial production method that uses machinery and a systematic approach to manufacturing goods, centralizing workers and machines in a single location.

Market Revolution

A dramatic change in the U.S. economy in the early 19th century, characterized by rapid industrialization, the expansion of internal markets, and the increase in transportation networks.

African-Americans

An ethnic or racial category in the USA comprised of individuals who have ancestral ties to any of the African black racial groups.