Problem 1: For the following linear programming problem: ???????? ? = 40?1 + 50?2

Subject to constraints:

3?1 − 6?2 ≥ 30

?1 – 15 ≤ 3?2 2

?1 + 3 ?2 = 24

?1, ?2 ≥ 0

1- Find the optimal solution using graphical solution corner points method or iso profit line method. Please, show the values for state variable, decisions variables, and slack and surplus variables

2- Determine the value for basic solution and non-basic solution, binding constraints and nonbinding constrains, and if there are any redundant constraints

3- Identify if there is any special case solution and state it.

solve using linear programming graphical solution

A mass of 50 g stretches a spring 3.828125 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 50 cms, and if there is no damping, determine the position u of the mass at any time t.

Enclose arguments of functions in parentheses. For example, sin(2x).

Assume g=9.8 ms2. Enter an exact answer.

u(t)=     m

When does the mass first return to its equilibrium position?

Enter an exact answer.

t=     s

James Bond, Q, and M have agreed to meet at a pub after work for drinks. Bond cannot remember if they agreed to meet at the “Fanny on the Hill” or at “My Father’s Moustache” - so he tosses a coin to decide which pub to go to. Q is also in the same predicament; he tosses a coin to decide between “My Father’s Moustache” and “The Quiet Woman”. M faced with same quandary flips a coin first to decide whether or not he needs to head to the “Fanny on the Hill”. If the answer is “no”, then he flips again to decide between “My Father’s Moustache” and “The Quiet Woman”. What is the probability that

(a) Bond and Q meet? (b) Q and M meet? (c) all three meet? (d) all three go to different places? (e) at least two meet?

Write a program (fortran 90) that calls a subroutine to approximate the derivative of y=sin(x)+2x^2 using a one-sided difference approach fx = (fi-fi-1)/deltaX and a centered difference approach fx = (fi+1-fi-1)/deltaX. The value of the function f and its derivative fx should be evaluated at x=3.75. Your code should print both values tot he screen when it runs.

Express sin 6θ as a polynomial in sin θ and cos θ.

Find the general solution of the differential equation y′′+9y=13sec²(3t), 0<t<π/6.

Use C1, C2,... for the constants of integration. Enter an exact answer. Enter ln|a| as ln(|a|), and do not simplify.

y(t)=

Use the method of variation of parameters to find a particular solution of the differential equation

4 y′′−4 y′+y=32et²

Find the inverse Laplace transform for

1 / ((s^2+1)(s+1))

How would you go about finding which of the following PDEs can be separated into ODEs by assuming a product of unknown functions x,y, and z?

1) aUxy+bU=0

2) x^2Uxx+yUyy=0

3) Uxx+3Uxy+7Uy=0

4)Uxx+Uyy+Ux+Uy=0

Where y=y(x) solves...

(5.9y-(23.6)x)dx+(5.9x-1)dy=0   y(0.27)=2.15

(5xy-(-1.1)ye^x -1)dx+(2.5x^2-(-1.1)e^x)dy=0     y(1.2)=2.3

The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Price in Dollars 21 26 28 35 43 Number of Bids 1 3 5 6 9 Table Step 6 of 6 : Find the value of the coefficient of determination. Round your answer to three decimal places.

4. The check digit for ISBNs is one of the numbers 0, 1, 2, . . . , 9, or the letter X. One of the your fellow students comments ”Gee, it sure is a pain to have to use that X all time. Why don’t they just compute the check digit sum modulo 10 instead of modulo 11, so that we can get rid of the X?” Would this plan work? Prove your answer.

Assume that all ISBNs are 10 digits with the tenth digit the check digit. I know that using modulo 10 to compute the check digit will catch single digit errors, but it will not always catch errors regarding transposition of numbers in the ISBN. I am having a difficult time proving that the transposition errors will not always be caught.

Let (X, d_X) and (Y, d_Y) be metric spaces.Define the function

d : (X × Y ) × (X × Y ) → R

by

d ((x₁, y₁), (x₂,y₂)) = d_x(x₁,x₂)+d_y(y₁,y₂)

Prove that d is a metric on X × Y .

Consider the differential equation y' = -18 + 11x - x².

(a) Find the equilibria (i.e. constant solutions).

(b) By analyzing the sign of dy/dt around the equilibria, determine whether each equilibrium solution is stable, unstable, or neither.

(c) Find an explicit general solution to the equation.

(d) Graph the equilibrium solutions, as well as some non-constant solution curves, verifying visually the stability properties determined in (b).

Consider the following relation

AIRLINE TABLE

Some of the requirements that this table is based on are as follows:

1. The AIRLINE is flying number of flights on various routes daily.
2. A flight with a particular Flight Id always flies on the same route (same Origin and same Destination).
3. Every captain has a unique Captain Id, and a non-unique Captain Name.
4. For each instance of a flight (a particular flight on a particular Flight Date) we keep track of who was the captain of that instance of a flight and how many passengers were on board (Number of Passengers on the Flight).
5. A captain can fly multiple flights during the same day.

Normalize the relation to the second normal form (2NF).

Normalize the relation to the third normal form (3NF).