1. Cost Estimation; High-Low and Regression Methods The Mac Davis Company specializes in the purchase, renovation, and resale of older homes. Mac employs several carpenters and painters to do the work for him. It is essential for him to have accurate cost estimates so he can determine total renovation costs before he purchases a piece of property. If estimated renovation costs plus the purchase price of a house are higher than the house’s estimated resale value, it is not a worthwhile investment.
   
   Mac has been using the home’s interior square feet for his exterior paint cost estimations. Recently he decided to include the number of external openings—the total number of doors and windows in a house—as a cost driver. Their cost is significant because they require time-consuming preparatory work and careful brushwork. The rest of the house usually is painted either by rollers or spray guns, which are relatively efficient ways to apply paint to a large area. Mac has kept careful records of these exterior painting costs on his last 12 jobs:

Required

1. Using the high-low cost estimation technique and square feet as the independent variable, determine the cost of painting a 3,300-square-foot house with 14 external openings. Also determine the cost for a 2,400-square-foot house with 8 externals openings.

2. Repeat requirement 1, but use number of external openings as the independent variable.

3. Plot the cost data against square feet and against openings. Which variable is a better cost driver? Why?

4. Create a multiple regression model for predicting cost based on openings and square feet. Comment on the statistical reliability and precision of this model.

5. What are the sustainability issues for this company, and what is the role of cost estimation in this regard?

Abstract Algebra

Let G be a discrete group of isometries of R2.

Prove there is a point p ∈ R2 whose stabilizer is trivial.

Let P₂ be the vector space of all polynomials of degree less than or equal to 2.

(i) Show that {x + 1, x² + x, x − 1} is a basis for P₂.

(ii) Define a transformation L from P₂ into P₂ by: L(f) = (xf)'    . In other words, L acts on the polynomial f(x) by first multiplying the function by x, then differentiating. The result is another polynomial in P₂. Prove that L is a linear transformation.

(iii) Compute the matrix representation of the linear transformation L above with respect to the basis for P₂ from the first part of this problem.

1. Find the Legendre polynomial PL(x) for L = 3,4,5,6 where the polynomian is the series solution for Legendre equation

2. Find the other solution QL(x) for the Legendre equation for L = 0,1,2

Please explain in full.

6) (a) Denote the successive intervals that arise in the bisection method by [a₀,b₀, [a₁,b₁] , [a₂,b₂], and so on

   Show that   b_n – a_n = 2^-n(b₀-a₀)

(b) The bisection method is said to have a linear convergence. Explain as clearly as possible what that means

c) For the bisection method, prove that |c_n – c_n+1| = 2^-n-2 (b₀ – a₀)

   where cn is the midpoint of each interval (ie) c_n = (a_n + b_n) /2

This is your lucky day. You have won a $20,000 prize. You are setting aside $8,000 for taxes and partying expenses, but you have decided to invest the other $12,000. Upon hearing the news, two different friends have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve spending some of your time next summer as well as putting up cash. Becoming a full partner in the first friend’s venture would require an investment of $10,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be $9,000. The corresponding figures for the second friend’s venture are $8,000 and 500 hours, with an estimated profit to you of $9,000. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this fraction. Because you were looking for an interesting summer job anyway (maximum 600 hours), you have decided to participate in one or both friends’ ventures in whichever combination that would maximize your total estimated profit. You need to solve the problem of finding the best combination.

a) Use the graphical solution method to solve this problem. Clearly display the feasible region of the problem and its optimal solution.

b) What profit would the first friend have to offer you in order to be optimal to invest your money and time to become his full partner?

Prove that if a finite group G has a unique subgroup of order m for each divisor m of the order n of G then G is cyclic.

The problem is for engineering mathematics,thanks.

Find all solutions.

sin z = 100

Use the Euler method to solve the following differential equation for the domain [2,2.5]. Use the step-size ℎ = 0.1. ?′=?ln?/? ;?(2)=?¹
b) Use the third order Taylor series method to find ?(0.1) and ?(0.2),
where ?′=1+2?? ;?(0)=0. Use the step-size ℎ=0.1.
c) Solve the problem in part (ii) using the fourth order Runge – Kutta method.
d) Solve the problem in part (ii) using the Predictor – Corrector method.

a building supply store prices all products to give a one third (33.33%)margin.
a.what rate of markup do they use?
b.if the company had profits of$500,000 what was their cost of goods sold?
c.if operating expenses are 10% of sales what is the percent net profit?
d.a mitre saw had a cost of$100 what was the selling price?
e.a pressure washer which costs $120 was sold after being marked down 20% what was the selling price ?what was the percent margin.

Write up a full proof of the fact that every k-dimensional subspace of R^n is the intersection of (n-k) hyperplanes. Tip: If you don't know how to start, begin by summarizing your answers to the previous problems on this lab.

In general, what do you need to show to prove the following?: (For example: to prove something is a group you'd show closure, associative, identity, and invertibility)

a. Ring

b. Subring

c. Automorphism of rings

d. Ring homomorphism

e. Integral domain

f. Ideal

g. Irreducible

h. isomorphic

the result of a recent bc election are summarised by party below

liberal      1,140,000
Nfp           430,000
green    250,000
reform     180,000
a. in a system of proportional representation (each party would receive seats in proportikn yo fhe number of votes)how many of the 79 seats would each partybreceive?
b.in the next election there were again 2 million votes cast.the green and reform parties received the same amount of votes as beforevbut the ndp received half as many votes as the liberals.how many seats would each partyvreceive?