A floor-refinishing company charges $1.83 per square foot to strip and refinish a tile floor for up to 1000 square feet. There is an additional charge of $350 for any job over 150 square feet.

a) Create a piecewise model to express the cost, C, of refinishing a floor as a function of the number of square feet, s, to be refinished.
b) Graph the function. Be sure to label your axis and use an appropriate scale.
c) Give the domain and range.

Will Rogers spun a lasso in a vertical circle. The diameter of the loop was 6 ft, and the loop spun 50 times each minutes. If the lowest point on the rope was 6 inches above the ground, write an equation to describe the height of this point above the ground after t seconds.

Please write nicely.

Make a code in matlab to know the determinant of a matrix n x n, using the sarrus rule.

Maximize or minimize the following functions. Be sure to check your second-order conditions.

(a) maxQ Π = (20 − Q)Q − 4Q

(b) maxQ Π = (12 − Q)Q − 2Q2

(c) maxL Π = AL1/2 − 4L

(d) maxL Π = ALα − wL (where A and α are exogenous parameters and w is the wage rate on labor, L)

For the following functions are they concave or convex and what does that depend on? If the answer is possibly both, over what ranges are the functions convex or concave?

(a) Y = AKαL 1−α with respect to K? with respect to L?

(b) Y = (αKρ + βLρ ) 1/ρ with respect to K? with respect to L?

(c) T C = 3Q3 + 2Q2 − Q + 10

The population of Americans age 55 and older as a percentage of the total population is approximated by the function f(t) = 10.72(0.9t + 10)0.3 (0 ≤ t ≤ 20) where t is measured in years, with t = 0 corresponding to the year 2000.† (Round your answers to one decimal place.) At what rate was the percentage of Americans age 55 and older changing at the beginning of 2003? % per year At what rate will the percentage of Americans age 55 and older be changing in 2018? % per year What will be the percentage of the population of Americans age 55 and older in 2018? %

7. (16 pts) a. Show that 11 is a primitive root of 13. b. What is the discrete logarithm of 4 base 11 (with prime modulus 13)?

Question 3

Briefly discuss the implications of the Capital Asset Pricing Model for the relationship between the current spot price of an asset and the discount offered by the seller of a futures contract. (100 words)

On April 11, 2014, Cynthia received a loan of $45,000 at 5.55% compounded monthly. On May 13, 2015, the interest rate on

the loan changed to 5.75% compounded quarterly and remained constant thereafter. What will be the accumulated value

of the loan on December 31, 2017?

1. What is the function and argument to calculate the sum of the values in the range D1:D50 for which in the adjacent value in the range A1:A50 equals “Senior” and the adjacent value in the range B1:B50 equals “B”?
2. What is the function and argument to calculate the average of the cells in the range C1:C50 for which the adjacent cell in the range B1:B50 equals “B”?
3. What is the function and argument to return the index number of the cell within the range E1:E50 that is equal to “Carter”, using an exact match?

Exercise 3
Part 1. Solving a system Ax = b **Create a function in MATLAB that begins with: function [C,N]=solvesys(A) [~,n]=size(A); b=fix(10*rand(n,1)) format long We are using format long to display a number in exponent format with 15 digit mantissas. If later on, you will need to switch back to the default format, type and run format The input is an matrix A. If A is invertible, the outputs are the matrix C, whose 3 columns x1, x2, x3 are the solutions obtained by the three methods described above, respectively, and N is the vector whose 3 entries are the 2-norms of the vectors of the differences between each two solutions.

**First, check if A is invertible. If A is not invertible, the function returns an empty matrix C=[ ] and an empty vector N=[ ]. Also, in this case, there are two possibilities for the system Ax = b: either “the system is inconsistent” or “the solution is not unique”. Use the command rank to check it on the two possible cases and program the two corresponding output messages. After that, terminate the program.

If A is invertible, the function solves the equation Ax = b using the three methods described above and gives the output vectors x1, x2, x3 for each of the methods (1)-(3), respectively. The vectors have to be the columns of the matrix C. Thus, we assign: C=[x1, x2, x3 ];

**The function [C,N]=solvesys(A) also returns a column vector N=[n1;n2;n3]; where n1=norm(x1-x2); n2=norm(x2-x3); n3=norm(x3-x1); The entries of the vector N are the 2-norms of the vectors of the differences between each two distinct solutions. Each entry is calculated by using a built-in MATLAB function norm, which is a square root of the sum of squares of the entries of the vector. The vector N gives an idea how “different” are the solutions obtained by various methods.

**Type the function solvesys in your Live Script.

**Run the function [C,N]=solvesys(A) for the following choices of the matrix A:

(a) A = magic(6); (b) A = magic(7); (c) A = eye(4); % Write a comment on the output for part (c) by comparing the solution with the vector b. (d) A = randi(20,4), (e) A = magic(3); (f) A = hilb(7)

Part 2. Condition numbers **Find the condition numbers of the matrices A=magic(7) and A=hilb(7) c1=cond(magic(7)) c2=cond(hilb(7))

% Compare c1 and c2 with number 1 and explain in your dairy file why the norms of the differences between the solutions for the coefficient matrix in part (f) are so big compared with the ones for the matrix in part (b).

**Explore the sensitivity of a badly conditioned matrix hilb(7): Input: A=hilb(7); Run the following: b=ones(7,1); x=A\b; b1=b+0.01; y=A\b1; norm(x-y) c3=rcond(A) %Using the output c3, which is the reciprocal condition number, explain why the system with the coefficient matrix hilb(7) is sensitive to perturbations.

**Re-run the code above for: A=magic(7);

%Comment on sensitivity to perturbations of magic(7) compared with hilb(7) by analyzing the corresponding outputs for the norm(x-y) and c3.

Problem 2: Indirect and Euclidean proofs (40 pts) For the following problems, you must use an indirect proof technique.

(a) (10 pts) Prove indirectly that, if a 2 is a multiple of 31, then so is a. Your proof should not consist of 30 cases – this includes absolutely no implied cases using horizontal dots (· · ·) and/or vertical dots (. . .).

(b) (15 pts) Using the result of question (a), prove that √ 31 is not a rational Q using the Euclidean method.

(c) (15 pts) Using the result of question (a), prove that √ 31 is not a rational Q using the Unique Prime Factorization Theorem.