. As part of his summer job at a resturant, Jim learned to cook up a big pot of soup late at night, just before closing time, so that there would be plenty of soup to feed customers the next day. He also found out that, while refrigeration was essential to preserve the soup overnight, the soup was too hot to be put directly into the fridge when it was ready. (The soup had just boiled at 100 degrees C, and the fridge was not powerful enough to accomodate a big pot of soup if it was any warmer than 20 degrees C). Jim discovered that by cooling the pot in a sink full of cold water, (kept running, so that its temperature was roughly constant at 5 degrees C) and stirring occasionally, he could bring the temperature of the soup to 60 degrees C in ten minutes. How long before closing time should the soup be ready so that Jim could put it in the fridge and leave on time ?

Let [x, t] denote the temperature of the soup at time t. The initial tempearture of the soup is [x, 0]=100 degrees C. The ambient temperature, i.e., the temperature of the sink of cold water is 5 deggrees C. According to Newton's law of cooling, the evolution of [x, t] is governed by the following differential equation:

               dx/dt=-k ([x, t]-5),

[x, 0]=100.

where k>0 is a constant that characterizes the cooling process. This differential equation asserts that the rate of change of the temperature of the soup is negatively proportional to the difference in its temperature and the temperature of the ambient environment. That is, the greater the difference between the temperature of the soup and the sink of cold water, the faster the soup cools.

(a) Solve this differential equation.

(b) We are told that [x, 10]=60 degrees C. Use this information to compute the parameter k.

(c) Find the time t at which [x, t]=20 degrees C.

Use the Laplace transform to solve the given initial-value problem.

y'' + 6y' + 34y = δ(t − π) + δ(t − 3π),    y(0) = 1,  y'(0) = 0

y(t) = ( ) + ( ) * U( t - pi ) + ( ) * U( t - )

The total number of people infected with a virus often grows like a logistic curve. Suppose that 20 people originally have the virus, and that in the early stages of the virus (with time, t, measured in weeks), the number of people infected is increasing exponentially with r=1.8. It is estimated that, in the long run, approximately 6250 people become infected.

(a) Use this information to write down a logistic differential equation for population P

dP/dt=

Then find the solution:

P=

(b) Sketch a graph of your solution. Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point? vertical coordinate =

(In practice, it may be easier to plot your answer for dP/dt instead, and pick the value of P for which it is maximal.)

Suppose G = Z_2 x Z_4. Find the normal subgroups where N is a normal subgroup of G and H is a normal subgroup of G

s.t. N is isomorphic to H but G/N is not isomorphic to G/H.

2. (CFGs) Give CFGs that generate the following languages. Use as few variables as you can (and no more than requested). Unless specified otherwise, the alphabet is Σ = {0, 1}.

(c) L3 = {w | w represents a binary number that starts with 1 and is divisible by 3}. Use at most 4 variables.

(d) L4 = {x₁#x₂# · · · #x_k | k ≥ 1, each x_i ∈ {0, 1}^* , and x_i = x_j^R for some i != j}. Recall that w^R represents string w written backwards. Use at most 4 variables.

1. A company produces three combinations of mixed vegetables that sell in​ 1-kg packages. Italian style combines 0.4 kg of​ zucchini, 0.4 kg of​ broccoli, and 0.2 kg of carrots. French style combines 0.5 kg of broccoli and 0.5 kg of carrots. Oriental style combines 0.2 kg of​ zucchini, 0.3 kg of​ broccoli, and 0.5 kg of carrots. The company has a stock of 18,000 kg of​ zucchini, 30,200 kg of​ broccoli, and 36,800 kg of carrots. How many packages of each style should it prepare to use up existing​ supplies?

2. A knitting shop orders yarn from three suppliers in​ Toronto, Montreal, and Ottawa. One month the shop ordered a total of 108 units of yarn from these suppliers. The delivery costs were ​$82,​ $45​, and ​$67 per unit for the orders from​ Toronto, Montreal, and​ Ottawa, respectively, with total delivery costs of ​$6453. The shop ordered the same amount from Toronto and Ottawa. How many units were ordered from each​ supplier?

3. Use the​ Gauss-Jordan method to solve the following system of equations.

Show that any two permutations σ,τ ∈ Sn have the same cycle structure if and only if there exists γ ∈ Sn such that σ = γτγ^−1.

Write a script or function in Python that approximates the solution to the system ??⃗=?⃗⃗ using the Jacobi Method. The inputs should be an nxn matrix A, an n-dimensional vector?⃗⃗, a starting vector ?⃗0,an error tolerance ?, and a maximum number of iterations N. The outputs should be either an approximate solution to the system??⃗=?⃗⃗ or an error message, along with the number of iterations completed. Additionally, it would be wise to build in functionality that allows you to optionally print the current estimated solution value at each iteration.

Solve the differential equation by variation of parameters. y′′ + 3y′ + 2y = 1/(7 + e^x)

y(x) =

1. Find the equation of the line with the given description: Horizontal, passes through (0,−8)(0,−8). (Use symbolic notation and fractions where needed.)

y=

2. Find the equation of the line with the given description: Passes through (−1,6)(−1,6) and (12,10)(12,10). (Use symbolic notation and fractions where needed.)

y=

3.Find the equation of the vertical line, passes through (76,89).(76,89). (Use symbolic notation and fractions where needed.)

x=

4.Determine whether there exists a constant cc such that the line x+cy=−3x+cy=−3

Has slope −8−8 =

Passes through (10,5)(10,5) =

Is horizontal =

Is vertical =

Note: In each case, your answer is either the value of cc satisfying the requirement, or DNE when such a constant cc does not exist.

How does Walmart uses breakeven analysis?

The manager of a seafood restaurant was asked to establish a pricing policy on lobster dinners. The manager intends to use the pricing $/LB to predict the lobster sales on each day. The pertinent historical data are collected as shown in the table. Anaswer the following questions.

  
a) x = independent variable. According to this problem, the ∑x =  

b) r i s the coeefficient of correlation. Use the r equation to compute the value of the denominator part of the equation. The value for the rdenominator =  (in 4 decimal places)

c) According to this problem, the correlation of coefficient, r, between the two most pertinent variables is =  (in 4 decimal places).

d) According to the instructor's lecture, the correlation strength between any two variables can be described as strong , weak , or nocorrelation. The correlation strength for this problem can be described as  correlation.

e) According to the instructor's lecture, the correlation direction between any two variables can be described as direct or indirectrelationship. The correlation direction for this problem can be described as  relationship.

f) Regardless, you were told to use the Associative Forecasting method to predict the expected lobster sale. If the lobster price = $8.58, the expected #s of lobster sold =  (round to the next whole #).

Let f(x)= kx^k-x^(k-1)-x^(k-2)-...-x-1, where k is an integer greater than or equal to 1. Prove the roots of f have absolute value less than or equal to 1.

This is possibly using Cauchy's Estimates for Roots of Polynomials or Ostrowski's Theorem, but I'm not sure how to use them.

Talia and Dina are planning to meet at he YMCA, which lies 5km North of the City Hall. At 3pm, Talia heads in a straight line with constant speed toward the YMCA from a point 1km South and 3km West of the City Hall. After 5 minutes she is 2.5km West, 0km South of the City Hall

a) Let the City Hall be the origin of a coordinate system. Find the parametric equations for Talia’s route.

b)Dina starts 3km East and 1km North of the City Hall on a straight line. Given that she walks at a speed of 5km per hour, at which time should she leave so that she and Talia arrive at the same time at the YMCA?

Prove that a disjoint union of any finite set and any countably infinite set is countably infinite.

Proof: Suppose A is any finite set, B is any countably infinite set, and A and B are disjoint. By definition of disjoint, A ∩ B = ∅

In case A = ∅, then A ∪ B = B, which is countably infinite by hypothesis.

Now suppose A ≠ ∅. Then there is a positive integer m so that A has m elements and there is a one-to-one correspondence f: {1, 2, 3, .., m} → A. In addition, since B is countably infinite, there is a one-to-one correspondence g: ℤ⁺ → B.

Let h: ℤ⁺ → A ∪ B be the function selected below. (Select one definition for h and use it for the rest of your answer.) We will show that h is one-to-one and onto.

Then h is one-to-one because f and g are one-to-one and A ∩ B = 0.  Further, h is onto because f and g are onto and given any element x in A ∪ B, x is in A or x is in B.

In case x is in A, then, since f is onto, there is an integer r in {1, 2, 3, ..., m} such that f(r) = x. Since r is in {1, 2, 3, ..., m}, r ≤ m, and so h(r) = _________.

In case x is in B, then, since g is onto, there is an integer s in ℤ⁺ such that g(s) = x.Let t = s + m.Then s = t − m. Also t > m+1, and thus h(t) = g(t-m)=g(s)=_________.

Therefore, h is a one-to-one correspondence from ℤ⁺ to A ∪ B, and so A ∪ B is countably infinite by definition of countably infinite.

Please write in bold letters where the ______ are.