You have three lines of training modules: Company Training (CT), On-line Training (OT), and Academic Training (AT). For each sold CT, you will receive $1,000 in revenue, while for each sold OT, you will receive $800 and for each AT, you will receive $700. Each module lasts for one month. To deliver the module CT, UQ-HDTC requires 100 hours of data scientist and computer programmer time. The module OT requires 300 hours of data scientist and 500 hours of computer programmer time, while AT requires 200 hours of data scientist and 100 hours of computer programmer time. Suppose you has purchased 1,000 hours of data scientists time and 800 hours worth of computer programmer time for each month. How many CT, OT, and AT modules you should sell per month, so as to maximize your revenue, given the constraints on data scientist and computer programmer time? Please form the problem as an LP problem and solve it using Tableu form of Simplex method.

(a) If V1,V2⊂V show that (V2^⊥)⊂(V1^⊥) implies V1⊂V2

(b) If V1,V2⊂V , show that (V1+V2)^⊥=(V1^⊥)∩(V2^⊥) where we write V1+V2 to be the subspace of V spanned by V1 and V2 .

The number of men and women wearing hats at a recent baseball game is recorded. The results are shown below. Hat No Hat Total Men 35 21 46 Women 10 32 42 Total 45 53 88 If one of these people at the baseball game is selected at random, find the probability that a) the person is a woman b) the person was wearing a hat. c) the person was wearing a hat, given that the person was a man. e) the person is a man, given that they were not wearing a hat.

In this lesson, you are learning about systems of equations and three methods for solving them: graphing, substitution, and addition. Compare and contrast the three methods by discussing the following:

Is one method easier than the others for all systems, or does it depend on the system?

If it depends on the system, how could you tell before you begin solving the system which solution method would be the most efficient?

How is it possible to tell by inspection (look at it) whether a system of linear equations has one, zero, or an infinite number of solutions?

A farmer has 360 acres of land on which to plant corn and wheat. She has $24,000 in resources to use for planting and tending the fields and storage facility sufficient to hold 18,000 bushels of the grain (in any combination). From past experience, she knows that it costs $120 / acre to grow corn and $60 / acre to grow wheat; also, the yield for the grain is 100 bushels / acre for corn and 40 bushels / acre for wheat. If the market price is $225 per acre for corn and $100 per acre for wheat, how many acres of each crop should she plant in order to maximize her revenue?

A.

1. Set up a linear programming problem, choosing variables, finding a formula for your objective function, and inequalities to represent the constraints.

2. You will need to decide a reasonable range for your variables, and then put in a column of values within that range for the x-variable in column A. Then you want to solve each constraint equation for y, and use that formula to get values in the "y for C1", etc., columns (B, C, D). Then graph the three constraint lines on one graph (as you did in Lab 1: open its Word document if you need refreshing on how to do this).

3. Shade in the feasible region.

4. Find the corners of the feasible region using goal seek to find intersections of lines, as you did in Lab 1.

5. Find in column H the values of f at the corners of the feasible region.

6. Determine the maximum revenue.

7. Finally, using new objective functions for when the prices of corn are at their highs and lows, answer the final question. This only involves computing new values of the objective function, not any new graphing or constraints.

x represents:

y represents:

Formula for objective function: f =

Constraint 1:

Constraint 2:

Constraint 3:

Corners of feasible region           f at corners

So how many acres of each crop should she plant?

Formulate a system of equations for the situation below and solve.

A manufacturer of women's blouses makes three types of blouses: sleeveless, short-sleeve, and long-sleeve. The time (in minutes) required by each department to produce a dozen blouses of each type is shown in the following table.

The cutting, sewing, and packaging departments have available a maximum of 87, 176, and 52 labor-hours, respectively, per day. How many dozens of each type of blouse can be produced each day if the plant is operated at full capacity?

Chapter 3.6, Problem 20E in Introduction to Linear Algebra (5th Edition)

Find the basis for the null space and the range of the given matrix. Then use Gram-Schmidt to obtain the orthagonal bases.

Compare the monthly payments and total loan costs for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs.

You need a $80,000 loan.

Option​ 1: a​ 30-year loan at an APR of 7.15%

Option​ 2: a​ 15-year loan at an APR of 6.75%

Find the monthly payment for each option.

The monthly payment for option 1 is $___

The monthly payment for option 2 is $___

Find the total amount paid for each option.

The total payment for option 1 is $_____

The total payment for option 2 is $___​(Use the answers from the previous step. Round to the nearest cent as​ needed.)

​

Prove that R>2r. R is circumradius and r is inradius of a triangle.

Derive Heron's formula.

g. Now let’s say you wait just 5 years before you start saving for retirement, how much will that cost you in interest? How about 10 years? How about just 1 year? (10 points) Now you need to consider if that is enough. If you live to be 90 years old, well above average, then from the time you retire, to the time you are 90, you will have to live on what you have in retirement (not including social security). So if you retired at 65, you will have another 25 years where your retirement funds have to last. h. Determine how much you will have to live on each year. Note, we are neither taking into account taxes nor inflation (which is about 2% a year). (5 points) Let’s look at this from the other direction then, supposing that you wanted to have $50,000 a year after retirement. i. How much would you need to have accumulated before retirement? (5 points) j. How much would you need to start investing each year, beginning right now, to accumulate this amount? A “short-cut” to doing this is to first compute the effective yield at your retirement age, then divide this amount into Part (i). This is the amount you well need to invest each year. (5 points) k. That was just using $50,000, how much would you want to have each year to live on? Dream big or reasonable depending on your occupation! Now using that value, repeat parts (i) and (j) again. You need to state what you would want to live on and it needs to be something besides $50,000. (10 points) Your answer to (k) would work, if you withdrew all of your retirement funds at once and divided it up. However, if you left the money in the account and let it draw interest, it is possible that the interest itself would be enough to live on, or at the very least if you had to withdraw some of the principle, the remaining portion would still continue to earn interest. Essentially, what you have found is the upper bound for the amount of money that you will need to invest each year to attain your financial goals. l. Finish by summarizing what you have learned in the entire project and consider setting a goal towards saving for retirement. (Your answer should be in complete sentences free of grammar, spelling, and punctuation mistakes.) This should be a paragraph not one sentence. (10 points)

1.38.4. The diagonal of a parallelogram and segments from any point on the diagonal to the vertices through which the diagonal does not pass divide the parallelogram into two pairs of equal triangles.

A. 271 and 516

1.  Find the greatest common divisor, d, of the two numbers from part A using the Euclidean algorithm. Show and explain all work.

2.  Find all solutions for the congruence ax ? d (mod b) where a and b are the integers from part A and d is the greatest common divisor from part A1. Show and explain all work.

Construct the 2 × 2 matrix for the linear transformations R 2 → R 2 defined by the following compositions. In each case, write down the matrix of each transformation, then multiply the matrices in the correct order.

(a) A dilation by a factor of 4, then a reflection across the x-axis.

(b) A counterclockwise rotation through π/2 , then a dilation by a factor of 1/2 .

(c) A reflection about the line x = y, then a rotation though an angle of π.

Please write a few sentences on the following two topics:

1. The system of coordinates is a "bridge" between algebra and geometry. Why? How? Etc....

2. Relations between straight lines and linear equations.

Please write neatly, and in detail. Thank you so much!

A $1000 par value 4% bond with semiannual coupons matures at the end of 10 years. The bond is callable at $1050 at the ends of years 4 through 6, at $1025 at the ends of years 7 through 9, and at $1000 at the end of year 10. Find the maximum price that an investor can pay and still be certain of a yield rate of 5% convertible semiannually.

ANSWER: 922.05