LaRosa Machine Shop (LMS) is studying where to locate its tool bin facility on the shop floor. The locations of the five production stations appear in figure shown below.

In an attempt to be fair to the workers in each of the production stations, management has decided to try to find the position of the tool bin that would minimize the sum of the distances from the tool bin to the five production stations. We define the following decision variables:

X = horizontal location of the tool bin

Y = vertical location of the tool bin

We may measure the straight line distance from a station to the tool bin located at (X,Y) by using Euclidean (straight-line) distance. For example, the distance from fabrication located at the coordinates (1,4) to the tool bin located at the coordinates (X,Y) is given by .

Find y as a function of x:

y'''-12y''+27y'=80e^x

y(0)=29

y'(0)=11

y''(0)=21

I found the roots to be r=0,3,9 and c1=224/9 c2=13/3 c3=-2/9

not sure what is wrong with the answer I'm entering.

2. The data of the velocity of a body is given as a function of time in the table below. Determine the value of the velocity at t=16 seconds using: (this is the table)

t(s) 0 15 18 22

v(t) 22 24 37 25

a) Newton’s divided difference first order polynomial method.

b) Newton’s divided difference second order polynomial method.

c) Newton’s divided difference third order polynomial method.

1. The table shows the performance of a selection of 100 stocks after one year. (Take S to be the set of all stocks represented in the table. If a stock stayed within 20% of its original value, it is classified as "unchanged".)

Use symbols to describe the event that an Internet stock did not increase.

How many elements are in the event?

2. The table shows the performance of a selection of 107 stocks after one year. (Take S to be the set of all stocks represented in the table. If a stock stayed within 20% of its original value, it is classified as "unchanged".)

Compute n(P ∪ N')._____

What does this number represent?

a. n(P ∪ N') is the number of stocks that were either not pharmaceutical stocks, or were not unchanged after a year (or both).

b.n(P ∪ N') is the number of stocks that were either pharmaceutical stocks, or were unchanged after a year (or both).    

c.n(P ∪ N') is the number of stocks that were pharmaceutical stocks and were not unchanged after a year.

d.n(P ∪ N') is the number of stocks that were either pharmaceutical stocks, or were not unchanged after a year (or both).

e. n(P ∪ N') is the number of stocks that were not pharmaceutical stocks and were unchanged after a year.

3. Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows 3; B: the numbers add to 4; C: at least one of the numbers is 1; and D: the numbers do not add to 9. Express the given event in symbols. HINT [See Example 5.]

Either the numbers add to 4, or they add to 9, or at least one of them is 1.

How many elements does it contain?

4. Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows 6; B: the numbers add to 9; C: at least one of the numbers is 6; and D: the numbers do not add to 12. Express the given event in symbols. HINT [See Example 5.]

The red die shows 6 and the numbers add to 9.

How many elements does it contain?

1) 14x + 35y = 91

2) 56x + 72y = 40

3) 24x + 138y =18

*linear diophantine equations

a) find different solutions for each equation

b) find the value (t), (t) being an arbitrary integer

c) find the general solution for every equation.

d) based on theorem 4.16 give the statement that gives the necessary and sufficient conditions (if and only if) for the equation ax + by = c to have a solution.

(hint) the linear equation ax+by=c has a solution if and only if ______________________________________________.

find the mean using excel

1,2,3,11,22,33

Verify all axioms that show that the set of second degree polynomials is a vector space. What is the Rank?

P2 = {p(x)P | p(x) = ax^2 + bx + c where a,b,c E R}

1. For a special case, show that the Improved Euler’s Method may be referred to as the Trapezoid Rule. Support this claim by writing and solving an example problem using both methods.       

2. The error associated with Euler's Method is proportional to step size. Euler's method will yield error-free prediction for a certain condition. What is the condition? Support the claim.

3. Use the Taylor Series to show that the Improved Euler’s Method and Midpoint Method have the same order of accuracy, which is greater than the order of accuracy for Euler’s Method.

4. Use the forward finite difference approximation to derive Euler’s formula.

There is a retailer called "The Coop", which carries a wide variety of products with MIT's name and logo: shirts, caps, keychains, pens... you name it! Anything from golf balls and teddy bears to pocket protectors and jewelry. With a store inside MIT's Student Center, and another one in the popular Kendall Square, across the street from MIT campus, The Coop is a favorite stop for casual MIT visitors, current students and their relatives, and nostalgic alumni.

The Coop carries a line of products that feature MIT's official seal cast in a jewelry-grade steel. The same seal (about the size of a coin) is used in multiple finished goods, such as necklaces, tie pins, cufflinks, and paperweights. The Coop is considering introducing in 2018 a new line of products featuring MIT's seal cast in 18 karat gold. These seals would be used in upscale jewelry, and as an ornament in the diploma frames that sell especially during commencement season.

Based on their experience with similar products in the past, The Coop has projected the following demand for gold MIT seals for each month of 2018.

The seals would be produced, under license, by Seventh Seal, a specialty manufacturer of commemorative seals cast in precious metals, located in Syracuse, New York. Seventh Seal has offered to produce the gold MIT seals exclusively for The Coop. Seventh Seal has explained to The Coop that there is a cost to setting up the equipment to produce the gold MIT seal. Because of this, Seventh Seal will charge The Coop a set-up cost of $1417.5 every time the MIT seal is produced.

Because of the high set-up cost, one of the managers at The Coop proposes doing all the seals that will be needed for 2018 in a one-time production batch. "This will save us a lot of money in set-up costs", he says.

Another manager, however, warns that - because the gold seals are expensive - there is an associated holding cost. The Coop estimates that the holding cost will be about $0.21000000000000002 per seal per month (the dot is a decimal mark).

Because of the high holding costs, that manager proposes doing monthly batches in the amount of seals that will be required that month, according to the demand projection. "This will save us a lot of money in holding costs", she says.

What are the total costs (e.g. the sum of set-up costs and holding costs) of producing the gold seals using a lot for lot (or chase) approach and a one time approach?

Show that {t_(1,s) : 2 ≤ s ≤ n} is a minimal generating set for S_n. You may use the fact that {t_(r,s) : 1 ≤ r < s ≤ n}, as defined in the outline, generates S_n.

Define the sample space. To get simpler expressions, you can prefer the set notations.
(a) (6 Points) What is the sample space of choosing a real number from interval [0, 1] which is larger than 0.5. Specify the properties of the sample space. Design two different random variables for this task.
(b) (6 Points) You play rock-paper-scissors with your friend. The one with total three wins or two subsequent wins will be the winner of the game. Give a probability tree of you being the winner. State the sample space by using probability tree diagram.
(You can use nested probability trees to make the representation easier. For example, you can generate a tree only for win-loss possibilities (without any information about rock-paper-scissors), one another tree for Win, including all possible cases with rock- paper-scissors, similarly one another tree for Loss, again including all possible cases with rock-paper-scissors)

This is a question for my problem-solving class. I am really stuck and I can't see much of a pattern so I would appreciate if someone could draw out for each thief and explain the pattern to get the answer for 40 thieves!

Question:

Forty thieves, all different ages, steal a huge pile of identical gold coins and must decide how to divide them up. They settle on the following procedure. The youngest divides the coins among the thieves however he wishes, then all 40 thieves vote on whether they are satisfied with the division. If at least half vote YES, the division is accepted. If a majority votes NO, the youngest is killed and the next youngest gets to try to divide the loot among the remaining 39 thieves (including herself). Again they all vote, with the same penalty if the majority votes NO and so on. Each of the thieves is logical and always acts in her or his own self-interest, ignoring the interest of the group, fairness, etc. Given all this, how should the youngest of the 40 thieves divide the loot?

Let L1 be the language of the binary representations of all positive integers divisible by 4. Let L2 be the language of the binary representations of all positive integers not divisible by 4. None of the elements of these languages have leading zeroes.

a) Write a regular expression denoting L1.

b) Write a regular expression denoting L2.

c) a) Draw a state diagram (= deterministic finite state automaton) with as few states as possible which recognizes L1. This state diagram should be complete: it should handle all strings of 0’s and 1’s.

d) Draw a state diagram (= deterministic finite state automaton) with as few states as possible which recognizes L2. This state diagram should be complete: it should handle all strings of 0’s and 1’s.

Gwen runs back and forth along a straight track. During the time interval 0≤t≤45 seconds, Gwen’s velocity, in feet per second, is modeled by the function v given by v(t)=(250sin(t^2/120))/(t+6).

(a) What is the first time, t1, that Gwen changes direction? Find Gwen’s average velocity over the time interval 0≤t≤t1.

(b) At time t=45 , how far is Gwen from her position at time t=0 ?

(c) Find the total distance that Gwen runs during the 45 seconds from time t=0 to time t=45 .

Q 3. The total mass of a variable density rod is given by

m=

where m = mass, ρx= density, Ac(x) = cross-sectional area, x = distance along the rod, and

L = the total length of the rod. The following data have been measured for a 20m length rod.

Determine the mass in grams to the best possible accuracy.

Q 4.With the aid of fourth order Runge-Kutta method, solve the competing species model

defined by

where the populations and are measured in thousands and t in years. Use a step size of

0.2 for 0 ≤ t ≤ 2 and plot the trajectories of the populations with Matlab or GNU Octave.