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Problem 462

Suppose a building contractor has accepted orders for 5 ranch style houses, 7 Cape Cod houses and 12 colonial style houses. The table below gives the amount of each raw material going into each type of house expressed in convenient units: 1) Represent the builder's order by a row vector. Represent the table above in matrix form. Find how much steel, wood, glass, paint and labor the building needs. 2) Suppose that steel costs \(\$ 15\) per unit, wood cost \(\$ 8\) per unit, glass costs \(\$ 5\) per unit, paint costs \(\$ 1\) per unit and labor costs \(\$ 10\) per unit. Represent this as a column vector and find the total cost for the builder.

Expert verified

In this exercise, we represent the builder's order as a row vector, \(\textbf{O} =\begin{pmatrix} 5 & 7 & 12 \end{pmatrix}\), and the raw material requirements for each house type as a matrix, \(\textbf{M}\). We multiply the order vector with the matrix to find the total amount of each raw material needed, \(\textbf{O}\textbf{M}\). Then, we represent the cost of each raw material as a column vector, \(\textbf{C}\), and multiply it with the total raw material requirements to find the total cost for the builder, \((\textbf{O}\textbf{M})\textbf{C}\).

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Chapter 19

1) Define sample space, random variable and expected value. 2) The probability that an American man in his early fifties will survive for another year is about \(.99\). How much should such a man pay for \(\$ 20,000\) worth of term insurance, exclusive of administrative costs and profit for the company? 3) Suppose that the probability of finding 0,1,2, and 3 people in line ahead of you at the Apex Supermarket is .2,4, .25, and .15, respectively. If the probability of finding 0,1,2 and 3 people in line at the B\&Q Supermarket across the street is 3,3, .2, and .2, respectively, at which supermarket would you find shorter lines in the long run? (Assume that both markets have the practice of opening up a new line as soon as one exceeds 3 people in length.)

Chapter 19

Consider a buyer who at the beginning of each month decides whether to buy brand 1 or brand 2 that month. Each month the buyer will select either brand 1 or 2 . The selection he makes at the beginning of any one month depends at most on the selection he made in the immediately preceding month and not on any other previous selections. Let \(\mathrm{p}_{\mathrm{ij}}\) denote the probability that at the beginning of any month the buyer selects brand \(j\) given that he bought brand i the preceding month. Suppose we know that \(p_{11}=3 / 4, p_{12}=1 / 4\), \(\mathrm{p}_{21}=1 / 2\) and $\mathrm{p}_{22}=1 / 2$ 1) First draw the transition diagram. 2) Assuming that the first month under consideration is January, determine the distribution of probabilities for April.

Chapter 19

Construct a simple model economy in which there are three industries: the crude oil industry, the refining industry that produces gasoline and the utility industry that supplies electricity. Suppose there are three types of consumers: the general public, the U.S. government and the export firms. The following chart gives the number of units that consumers need of crude oil, gasoline and electricity: The price of crude oil is \(\$ 4\) per unit, the price of gasoline is \(\$ 3\) per unit and the price of electricity is \(\$ 2\) per unit. 1) Find the demand vectors for the industries and consumers. 2) Find the total demand vector. 3) Find the price vector. 4) Find total costs for the industries. 5) Find the income vector. 6) Find the profits (or losses) of the three industries.

Chapter 19

1) Show that the standard deviation of \(\mathrm{X}\) and \(\mathrm{Y}\) corresponds to vector addition (according to the parallelogram law). 2) Solve the following problems: (a) Suppose that 3 resistors are placed in series. The standard deviations of the given resistances are 20,20 and 10 ohms, respectively. What is the standard deviation of the resistance of the series combination? (b) Gears \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}^{\prime}\) s widths have standard deviations of \(0.001,0.004\), and \(.002\) in., respectively. If these are assembled side by side on a single shaft what is the standard deviation of this total width?

Chapter 19

A certain textile mill finishes cotton cloth obtained from weaving mills. The mill turns out two styles of .cloth, a lightly printed style and a heavily printed one. The mill's output during a week is limited only by the capacity of its equipment for two of the finishing operations - - printing and bleaching .. and not by demand considerations. The maximum weekly output of the printing machinery is 800 thousand yards of cloth if the light pattern is printed exclusively, 400 thousand yards if the heavy pattern is printed exclusively, or any combination on the printing line $\mathrm{L}+2 \mathrm{H}=800\( (where \)\mathrm{L}$ represents light pattern and H heavy pattern). In a week, the maximum the bleaching equipment can handle is 500 thousand yards of the light-patterned cloth exclusively, 550 thousand yards of the heavy-patterned cloth exclusively, or any combination on the bleaching line, \(1.1 \mathrm{~L}+\mathrm{H}=550\). The mill gained \(\$ 300\) and \(\$ 290\) per thousand yards of the light . . and heavy-patterned cloths, respectively. 1) Draw the graph of the two lines described above. 2) Solve the linear programming problem of maximizing the gain from a week's production.

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