# Matrices, Systems of Equations, Linear Programming, Simplex Method and Measures of Central Tendency

Please see the attached file for the fully formatted problems.

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1.)

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2.) Find the solution of the following system of linear equations:

x + 2y - 7z = -1

3x + 7y - 24z = 6

x + 4y - 12z = 26

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3.) A cookie company makes three kinds of cookies, oatmeal raisin, chocolate chip, and shortbread, packaged in small, medium, and large boxes. The small box contains 1 dozen oatmeal raisin and 1 dozen chocolate chip; the medium box has 2 dozen oatmeal raisin, 1 dozen chocolate chip, and 1 dozen shortbread; the large box contains 2 dozen oatmeal raisin, 2 dozen chocolate chip, and 3 dozen shortbread. If you require 15 dozen oatmeal raisin, 10 dozen chocolate chip, and 11 dozen shortbread cookies, how many of each size box should you buy?

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4.) Find the inverse of:

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5.) The graph below shows the constraints of the objective function:

P = 3x + 2y

The shaded area is the set of all feasible points.

Using the graph above, find the maximum value of the objective function.

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6.) Use the Simplex Method to solve the following linear programming problem. Show all tableaus and make a notation of all row operations performed.

Maximize:

P = 2x1 + 8x2 + 10x3 + x4

subject to the constraints

x1 + 2x2 + x3 + x4 <= 50

3x1 + x2 + 2x3 + x4 <= 100

x1 >=0 x2 >= 0 x3 >= 0 x4 >= 0

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7.) Assuming your principal is $1000, which rate will yield the larger amount after 1 year?

a.) 9% compounded quarterly

b.) 9.25% compounded annually

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8.) What monthly payment is needed to pay off a loan of $500 amortized at 12% compounded monthly for 2 years?

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9.) A school board issues bonds in the amount of $15,000,000 to be retired in 20 years. How much must be paid into a sinking fund each year at 4% compounded annually to pay off the total amount due?

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10.) There are three sets:

A = { 1, 2, 3, 4 }

B = { 3, 5, 6, 7 }

C = { 1, 3, 7, 8 }

Find:

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11.) There are two small towns, Greenville and Yellowville. Greenville contains 20 houses and Yellowville contains 15 houses. The state safety inspector is asked to randomly inspect 5 houses in Greenville and 7 houses in Yellowville. In how many ways can this be done (Hint: this is not a permutation)?

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12.) The following table shows the weather conditions each day for the last 100 days:

Snowy Days Rainy Days Cloudy Days Sunny Days

5 20 40 35

Based on this data:

a.) What is the probability that tomorrow will be snowy?

b.) What is the probability that tomorrow will be rainy or cloudy?

c.) What is the probability that tomorrow will be rainy and the day after tomorrow will be sunny?

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13.) The probability of a newborn baby being a girl is 0.49. If four babies are born in a hospital on one day, what is the probability that all four are girls?

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14.) In 7 different rounds of golf, a very inconsistent golfer shot scores of:

120, 75, 95, 115, 82, 88, 99

a.) What is the arithmetic mean of the 7 scores?

b.) What is the standard deviation of the set of golf scores?

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15.) In the last month, ten houses sold in a local town. The price of each sale:

$150,000

$1,100,000

$130,000

$130,000

$170,000

$160,000

$200,000

$130,000

$115,000

$800,000

a.) What is the mean, median, and mode of this data?

b.) Which measure gives you a better sense of the midpoint of the housing prices in this area, the mean or the median? Why?

#### Solution Summary

Matrices, Systems of Equations, Linear Programming, Simplex Method and Measures of Central Tendency are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.