A company manufactures two products. For $1.00 worth of product B, the company spends $.45 on materials, $.25 on labor, and $.15 on overhead. For $1.00 worth of product C, the company spends $.40 on materials, $.30 on labor, and $.15 on overhead. Let b = [.45 .25 .15] and c = [.40 .30 .15] Then b

Question

A company manufactures two products. For $1.00 worth of product B, the company spends $.45 on materials, $.25 on labor, and $.15 on overhead. For $1.00 worth of product C, the company spends $.40 on materials, $.30 on labor, and $.15 on overhead. Let

[latex] b =\left [ \begin{matrix} .45 \ .25 \ .15 \end{matrix} ight ] [/latex] and [latex] c = \left [ \begin{matrix} .40 \ .30 \ .15 \end{matrix} ight ] [/latex]

Then b and c represent the “costs per dollar of income” for the two products.

a. What economic interpretation can be given to the vector 100b?
b. Suppose the company wishes to manufacture [latex] x_{1} [/latex] dollars worth of product B and [latex] x_{2} [/latex] dollars worth of product C. Give a vector that describes the various costs the company will have (for materials, labor, and overhead).

Accepted Answer

a. Compute

100b = 100[latex] \left [ \begin{matrix} .45 \ .25 \ .15 \end{matrix} ight ] [/latex] = [latex] \left [ \begin{matrix} 45 \ 25 \ 15 \end{matrix} ight ] [/latex]

The vector 100b lists the various costs for producing $100 worth of product B—namely, $45 for materials, $25 for labor, and $15 for overhead.

b. The costs of manufacturing [latex] x_{1} [/latex] dollars worth of B are given by the vector [latex] x_{1}b [/latex], and the costs of manufacturing [latex] x_{2} [/latex] dollars worth of C are given by [latex] x_{2}c [/latex]. Hence the total costs for both products are given by the vector [latex] x_{1}b + x_{2}c [/latex].

Question 1.3.7: A company manufactures two products. For $1.00 worth of prod...

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