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!

What is a Bessel Differential Equation? Classify Bessel Equation with respect to its order, linearity and homogeneity. How many independent solutions of Bessel Differential equation are needed to construct general solution?

One of the luxury car manufacturers usually wants to make a limited number of automobiles in a given year and the company’s aim is to expand their profit margins in each car sold. After doing the analysis on the sales of their cars, the company determined the below supply and demand functions of the cars sold:

Supply function: P = 1000 + Q^2

Demand function: P = 21000 – Q^2

a) Calculate the producer, consumer and total surplus at the market equilibrium.

b) If the manufacturer restricts the number of cars manufactured and sold to 800 at the demand price of $15,000, how much is the total producer surplus?

c) A manufacturer wants to maximize the producer surplus and hence determine the quantity (give the whole number) at which this can be possible. Also determine the producer surplus at this quantity.

(a) A bank’s safe has 5 dials each of which can be set to a number from 0 to 36 (so each dial has 37 possible settings).

i. How many different ways can the 5 dials be set?

ii. How many settings are there if no two dials may be set to the same number?

iii. How many possible settings are there if all the numbers must be even (count 0 as even), and duplicate numbers are permitted?

iv. Re-do the previous question, but this time with no duplicates allowed.

(b) Suppose the bank gets a new safe also with 5 dials, but each dial is numbered as follows: Dial 1 goes from 0 to 10 Dial 2 goes from 0 to 12 Dial 3 goes from 0 to 20 Dial 4 goes from 0 to 25 Dial 5 goes from 0 to 35

i. How many different ways can the 5 dials be set?

ii. How many settings are there if no two dials may be set to the same number?

iii. How many possible settings are there if all the numbers must be even (count 0 as even), and duplicate numbers are permitted?

iv. Re-do the previous question, but this time with no duplicates allowed.

Find the general solution of the given differential equation, and use it to determine how solutions behave as t→∞.

9y′+y=5t2

Find a power series solution of this ODE:

y''+x^2y'=0 ; y(0)=1 y'(0)= 2

Prove verbally without referencing a Venn diagram, can use a Vennn diagram to visually see it but cannot be a reference in proof.

a) Let A be a set. Prove that A⊆ A.

b) Let A, B, and U be sets so that A ⊆ U and B ⊆ U. Prove that A ⊆ B if and only if (U \ B) ⊆ (U \ A)

c) Let A, B be sets. Prove that A \ (A ∩ B) = A \ B

d) Let A, B be sets. Prove that A \ (B \ A) = A

1)

A full rectangular water tank has a base of 10 ft x 15 ft and is 20 ft high. What is the pressure at the bottom of the tank in a) PSI b) Pa

2)

A cargo ship has a volume of 10,000 cubic meters. The total mass of the ship and cargo is 8 million kg. Will it sink or float at the surface of seawater having a density 1025 kg/m3? Why?

what would be some practical research scenarios in the field of Social Work where this type of statistical test would be appropriate.

Prove that if f, g are integrable, then the function (f(x) + cos(x)g(x))2is integrable.