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?

Kane Manufacturing has a division that produces two models of hibachis, model A and model B. To produce each model A hibachi requires 3 lb of cast iron and 6 min of labor. To produce each model B hibachi requires 4 lb of cast iron and 3 min of labor. The profit for each model A hibachi is $3, and the profit for each model B hibachi is $2.50. If 1000 lb of cast iron and 20 labor-hours are available for the production of hibachis each week, how many hibachis of each model should the division produce each week to maximize Kane's profit?

What is the largest profit the company can realize?
$

Is there any raw material left over? (If so, give the amount remaining. If not, enter 0.)

Question 2. True or false

2. If [T]β is the matrix representing the linear map T in the basis β, then the jth column of [T]β contain the coordinates of the T(βj) in the basis β, and same for the rows.

1. Your firm designs training materials for computer training classes, and you have just received a request to bid on a contract to produce a complete set of training manuals for an 8-session class. From previous experience, you know that your firm follows an 85% learning rate. For this contract, it appears that the effort will be substantial, running 50 hours for the first session. Your firm bills at the rate of $100/hour and the overhead is expected to run a fixed $600 per session. The customer will pay you a flat fixed rate per session (Per Session Price.) If your profit markup is 20%, what will be the Total Price, the Per Session Price, and at what session will you break even?

Answer the following three questions:

1. What is the Total Price? This is what you would charge the customer so that you can have your profit markup of 20% over all of your costs. To calculate this, first figure out your cost per each session, add them up, and then add your profit.
2. What is the Per Session Price? This is the revenue that the customer pays you each time you complete a session. It is calculated by dividing the Total Price by the number of sessions.
3. What is the Break Even Point? At the beginning, your cost per session is more than your revenue per session. Gradually, your cumulative revenue matches the cumulative cost, and eventually exceeds it so that you can end up with the desired profit. The break-even point is the session at which, for the first time, your revenue exceeds your cost.

2. A manufacturing firm has set up a project for developing a new machine for one of its production lines. The most likely estimated cost of the project itself is $1 million, but the most optimistic estimate is $900,000 while the pessimists predict a project cost of $1,200,000. The real problem is that even if the project costs are within those limits, if the project itself plus its implementation cost exceed 1,425,000, the project will not meet the firm’s NPV hurdle. There are four cost categories involved in adding the prospective new machine to the production line: (1) engineering labor cost, (2) non-engineering labor cost, (3) assorted materials cost, and (4) production line down-time cost.

The engineering labor requirement has been estimated to be 600 hours, plus or minus 15% at a cost of $80 per hour. The non-engineering labor requirement is estimated to be 1500 hrs., but could be as low as 1200 hrs. or as high as 2200 hrs. at a cost of $35 per hour. Assorted material may run as high as $155,000 or as low as $100,000 but is most likely to be about $135,000. The best guess of time lost on the production line is 110 hours, possibly as low as 105 hours and as high as 120 hours. The line contributes about $500 per hour to the firms profit and overhead. What is the probability that the new machine project will meet the firm’s NPV hurdle? Use Crystal Ball simulation to answer the question.

Let f be an irreducible polynomial of degree n over K, and let Σ be the splitting field for f over K. Show that [Σ : K] divides n!.

A number is a Universal Product Code (UPC) if its last digit agrees with the following computations:

• The sum of the odd position digits (not including the last) is M. That is we add the first digit to the third digit to the fifth digit etc.

• The sum of the even position digits (not including the last) is N. •

c = (3M + N)%10.

• If c = 0 then the check digit is 0.

• If c 6= 0, then the check digit is 10 − c.

(a) Check whenever 1928467 is a UPC.

(b) Suppose you are given a 7 digit number which is a UPC. Prove that if a mistake is made when scanning the number, causing one digit to be read incorrectly, then you will be able to tell that an error has been made. Hint: use results from question 2.

(c) Find an example of two 7-digit UPCs that have equal last digits, and disagree with each other at exactly two of the other digits numbers.

An example with number 1231242.

1) you add all odd-positioned digits except the last one:

M=1+3+2=6

2) add all even positioned digits not including the last one:

N=2+1+4=7

3) c=(3M+N)%10=(6*3+7)%10=5

4) the check digit is 10-5=5

So 1231242 is not an UPC.

However if we change the last digit to be 5, then it will be UPC. That is 1231245 is a UPC

Solve the equation CX + 5X = D respect to the matrix X.
C = [ -4 3 5 ]

   [ 2 0 6 ]

   [ 1 2 -3 ]

D = [ 1 3 4 ]

[ 2 1 1 ]

   [ 4 3 2 ]

Thanks!