Textbook problems

Topic covered

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$63A,64A,65$

Use matrix inverse methods to solve an

application problem

$5.$ An international mining company owns two mines, one at Voisey's Bay in Labrador, Canada and the other at Hawk Ridge, in British Columbia. The composition of the ore in each field is shown in the table below:

$\backslash$table$[[$Mine$,$Nickel $(\%)$ per ton,Copper $(\%)$ per ton$],[$Voisey$\text{'}$s Bay,$2,4],[$Hawk Ridge,$3,2]]$

A$)$ How many tons of ore from each mine should be used to obtain exactly $6$ tons of nickel and $8$ tons of copper? Please show your work for part A$.$

B$)$ How does the answer to part A change to obtain $7\mathrm{.}5$ tons of nickel and $7$ tons of copper?

By the way, copper and nickel are essential elements in the electronic systems of smart phones.

Section $5\mathrm{.}3$

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Theorem

True$/$false statements about linear

$1$ and $2$

programming, solution regions, and corner points

$6.$ A$)$ If the feasible $($solution$)$ region in a linear program problem is bounded, then the objective function can be maximized, but not minimized.

B$)$ If the optimal solution to a linear programming program exists, it must occur at one or more of the corner points of the feasible region.

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$49$A$,$ and the example

from the lecture notes

Linear programming application:

Maximizing profit

$7.$ South Shore Sails manufactures regular and competition sails. Each regular sail takes $2$ hours to cut and $4$ hours to sew. Each competition sail takes $3$ hours to cut and $10$ hours to sew. There are $150$ labor hours available in the cutting department and $380$ hours in the sewing department. If the profit on a regular sail is $$100$ and the profit on a competitive sail is $$200,$ how many sails of each type should be made to maximize the company's profit? What is the maximum profit?

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$51A,53,$ and the example

Linear programming application:

from the lecture notes

Minimizing cost

$8.$ A special diet for laboratory animals must contain at least $1,160$ units of vitamins and no more than $850$ calories. There are two feed mixes available, mixA and mix $B.$ One gram of Mix A contains $4$ units of vitamins and $2$ calories. One gram of mix $B$ contains $5$ units of vitamins and $5$ calories. If mix A costs $$0\mathrm{.}04$ per gram and mix $B$ costs $$0\mathrm{.}09$ per gram, how many grams of each mix should be used to meet the requirements of the diet at minimal cost? What is the minimum cost?

$($Answers on next page$)$