QUESTION 1 A company manufactures three products each of which requires certain amounts of three raw materials as well as labour. The matrix R summarizes the requirements per unit of each product Raw material Labour Product (1 2 3 6) A (2 3 2 8) B (4 2 5 4) C Raw material requirements are stated in pound per unit and labour requirements in hours per unit. The raw materials costs $2, $8 and $2.50 respectively. Labour costs are $8 per hour. Assume 800, 2000, and 600 units of product A, B, C are to be produced. Required i. Perform matrix multiplication where compute total quantities of the four resources required to produce the desired quantity of products A, B, and C. ii. Using your answer from part a, perform a matrix multiplication which calculates the combined total cost of production.
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Let $U$ be the $3 \times 2$ cost matrix described in Example 6 of Section $1.8 .$ The first column of $U$ lists the costs per dollar of output for manufacturing product $B,$ and the second column lists the costs per dollar of output for product $C .$ (The costs are categorized as materials, labor, and overhead.) Let $\mathbf{q}_{1}$ be a vector in $\mathbb{R}^{2}$ that lists the output (measured in dollars) of products $\mathrm{B}$ and $\mathrm{C}$ manufactured during the first quarter of the year, and let $\mathbf{q}_{2}, \mathbf{q}_{3},$ and $\mathbf{q}_{4}$ be the analogous vectors that list the amounts of products $\mathrm{B}$ and $\mathrm{C}$ manufactured in the second, third, and fourth quarters, respectively. Give an economic description of the data in the matrix $U Q,$ where $Q=\left[\begin{array}{llll}{\mathbf{q}_{1}} & {\mathbf{q}_{2}} & {\mathbf{q}_{3}} & {\mathbf{q}_{4}}\end{array}\right]$
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