Find the $80$th term of the arithmetic sequence with first term $-2$ and common difference $3$.

Resolve Problem 1 by using the simplex method.

Andre assures Jessica that investing in the five different subaccounts is a more sensible approach and will still yield a composite rate of return of approximately $7.25 \%$. Do you agree with Andre? How would you check his math to verify the rate of return?

The following exercises describe apportionment problems for situations besides the House of Representatives. Do the following: a. Find the apportionment by Hamilton's method. b. Find the apportionment by Jefferson's method. c. Find the apportionment by Webster's method. d. Find the apportionment by the Hill-Huntington method. e. Compare the results of the various methods. A chain of hardware stores is reapportioning its 25 managers among stores in four locations according to the stores' monthly gross sales. The sales at the four stores are as follows: Boulder, $\$ 2.5$ million; Denver, $\$ 7.6$ million; Broomfield, $\$ 3.9$ million; Ft. Collins, $\$ 5.5$ million.

In trying to estimate the height of a nearby building. you make the following observations. If you stand $15$ feet away from a light post, you can line up the top of the building with the top of the light post. Furthermore, you can easily determine that the top of the light post is $10$ feet above your head and that the building is 50 feet from your sighting position. What is the height of the building? Draw a good picture!

Write a verbal description of the type of linear programming problem that can be solved by the method indicated Below, Include the type of optimization, the number of variables, the type of constraints, and any restrictions on the coefficients and constants. Problem:- Big $M$ method

The human lung has approximately $300$ million nearly spherical air sacs (alveoli), each with a diameter of about $\frac{1}{3}$ millimeter. The key feature of the air sacs is their surface area, because on their surfaces gas is exchanged between the bloodstream and the air.

a. What is the total surface area of the air sacs? What is the total volume of the air sacs?

b. Suppose a single sphere were made that had the same volume as the total volume of the air sacs. What would be the radius and surface area of such a sphere? How would this surface area compare to that of the air sacs?

c. If a single sphere had the same surface area as the total surface area of the air sacs, what would be its radius? Based on your results, comment on the design of the human lung.

Suppose you are designing a cardboard box that must have a volume of $8$ cubic feet. The cost of the cardboard is $\$ 0.15$ per square foot. What is the most economical design for the box (the one that minimizes the cost), and how much will the material in each box cost?

Consider the following lengths of one side of a golden rectangle. Find the length of the other side. Notice that the other side could be either longer or shorter than the given side. Use the approximation $\phi \approx 1.62$ for your work.

$5.8$ meters

Andre suggests that Jessica consider a more balanced approach to spread out her risk by investing $\$ 1,000$ per year in each of five different subaccounts in a variable annuity. Andre shows Jessica a spreadsheet detailing the following rates of return for each of the five subaccounts for 30 years: $0 \% ; 3 \% ; 5 \% ; 8 \%$; and $12 \%$. Use Table 14-1 to compute the total value for investing $\$ 1,000$ annually in each of the five subaccounts for $30$ years.

Write a verbal description of the type of linear programming problem that can be solved by the method indicated Below, Include the type of optimization, the number of variables, the type of constraints, and any restrictions on the coefficients and constants. Problem:- Basic simplex method with slack variables

Each movement of Mozart's $19$ piano sonatas is clearly divided into two parts (the exposition and the development and recapitulation). In a paper called "The Golden Section and the Piano Sonatas of Mozart" (Mathematics Magazine, Vol. $68,$ No. $4, 1995$), John Putz gives the lengths (in measures) of the first part $(a)$ and the second part (b) of each movement. Some of the data are given below.

a. The ratio of the length of the whole movement to the length of the longer segment is $(a+b) / b$. Add a third column to the table and compute this ratio for the given data.

b. Briefly comment on how well the ratios that you computed in part (a) are approximated by $\phi$.

c. Read the article by John Putz. Do you believe that Mozart composed with the golden ratio in mind?

Recall that the Cantor set is formed by starting with a line segment and then successively removing the middle one-third of each segment in the current figure. If a ruler the length of the original line segment is used, it detects one element in the Cantor set because it can't "see" details smaller than itself. If the ruler is reduced in size by a factor of $R=3$, it finds two elements (only solid pieces of line, not holes, are measured). If the ruler is reduced in size by a factor of $R=9$, how many elements does it find? Based on these results, what is the fractal dimension of the Cantor set? Explain why this number is less than $1$.

Officially there is a long-run goal of free labour mobility throughout the European Economic Area, to which 30 nations currently belong. For the year 2011, let $c_{i j}$ denote the number of workers who moved their main place of work from nation $i$ to nation $j, i \neq j$. If, say, $i=25$ and $j=10$, then we write $c_{25,10}$ for $c_{i j}$. If $i=j$ we let $c_{i j}=0$. Explain the meaning of the sums: (a) $\sum_{j=1}^{30} c_{i j}$, (b) $\sum_{i=1}^{30} c_{i j}$.