Task 2

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The Story of Archimedes

About 2,200 years ago, one of the most famous mathematicians in history made a major discovery. The mathematician was Archimedes, who was presented with a problem by King Hieron II of Syracuse in Sicily.

The king had given a bar of gold to a jeweler and ordered him to make it into a crown. After receiving the crown, the king suspected that the jeweler had replaced some of the gold with an equal weight of a cheaper metal like silver and kept the remainder of the gold for himself. The king had no way to determine whether this was true, so he gave the crown to Archimedes and asked him to devise a way to find out.

At the time, Archimedes knew that gold was more dense than silver. So, if the volume of the crown was greater than the volume of a bar of gold, they would have proof that the jeweler had stolen some of the gold. However, Archimedes had no way to find the volume of an irregular shape like a crown.

Archimedes struggled with this problem for a long time. One day, he went to take a bath. As he lowered himself into the bath, he noticed that the water level rose. As he continued to lower himself into the water, the water level continued to rise. He was so excited at his discovery that he jumped out of the bath and, before he could remember to put his pants on, went running through the streets yelling “Eureka! Eureka!” which in Greek means “I’ve found it! I’ve found it!” He now had a way to measure the volume of the irregularly shaped crown.

In Archimedes’s case, the jeweler had indeed stolen gold from the king. This did not end well for the jeweler.

Part A

A key component to this story is that the jeweler deceptively replaced “an equal weight” (or equal mass) of gold with silver. How does this action result in increasing the volume of the crown? Explain using one or more equations.

Part B

What did Archimedes discover in the bath? Why was he so excited? Describe the discovery using geometric terms. What do you predict Archimedes did next with his new discoveries? Explain.

why can't we use bisection methods or newton's method for nonconvex functions? x^4+x^3-2x^2-2x especially for this function?

A Palindromic number is one that reads the same backwards and forwards. Write a MATLAB function (call it palin.m) that takes as input a positive integer, and returns 1 (true) if it is palindromic, 0 (false) if it is not.  

1.) The demand model p=0.02x+19 gives the price per model (in dollars per novel) p when novels are sold. The cost (in dollars) for publishing x novels is given by C(x)=4x+19.

a.)How many novels should be sold in order for revenue to be a maximum?

b.)What is the maximum profit?

c.) Find the average cost when 50 novels are sold.

Prove that for any real numbers a and b, there exists rational numbers x and y where y>0 such that a < x-y < x+y < b

Determine the general solutions for y" + 4y' + 4y = 8x^2 , y" + 4y' 3y = 32xe^x , and  y" + 4y' + 5y = 5x − 1.

Suppose the table gives the number N(t), measured in thousands, of minimally invasive cosmetic surgery procedures performed in the United States for various years t.

t N(t)(thousands)

(b) Construct a table of estimated values for N'(t). (Use a one-sided difference quotient with an adjacent point for the first and last values, and the average of two difference quotients with adjacent points for all other values. Round your answers to two decimal places.)

For my Hospitality Management class I need to convert a recipe that service 4 to 6 people as the Yield to 84 serving as the portions.

Do I divide 84 by 4 or 6???

Serving size of the recipe I'm converting is 4 oz of meat, and 4 oz vegetables.

Im not sure the conversion to make the recipe of 4 to 6 feed 84 people? What am I supposed to convert the entire recipe by??

Find a root of the following equation by using Newton's method in Matlab

9x^4 + 18x^3 + 38x^2 - 57x + 14 = 0;

The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin C daily. A cup of one dietary drink provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of another dietary drink provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. Suppose that the first dietary drink costs $0.12 per cup while the second dietary drink costs $0.15 per cup. Determine how many cups of each drink should be consumed each day to minimize the cost and still meet the stated daily requirements using the Standard Simplex Method.

Find rank, image, null space, eigenvalues, and eigenvectors

A= [1 2 0; 0 5 -1; 1 -3 1]

There are certain number of balls and they are painted with following conditions:
1. Every two colours appear on exactly one ball.
2. Every two balls have exactly one colour in common.
3. There are 4 colours such that any three of them appear on one ball.
4. Each ball has 3 colours.
Find number of balls and number of colours used.

(Answer: 7 balls and 7 colours)

A manufacturer of projection TVs must ship a total of at least 1000 TVs to its two central warehouses, x to the first warehouse and y to the second warehouse. Each warehouse can hold a maximum of 750 TVs. The first warehouse already has 150 TVs on hand, whereas the second has 50 TVs on hand. It costs $10 to ship a TV to the first warehouse, and it costs $15 to ship a TV to the second warehouse. How many TVs should be shipped to each warehouse to minimize cost?

What was the impact of the near failure of Bear Stearns and the failure of Lehman Brothers on Money Markets?

What actions did the Federal Reserve and the Treasury Department take? What were the impacts of the decisions if any?

.Consider the following continuous-time model of the dynamics of an epidemic system made of a growing susceptible population (x) and an infected population(y). Populations are nonnegative (i.e., x≥0, y≥0),and all the parameters are positive (i.e.,a> 0, b> 0, c> 0). dx/dt = ax ( 1 - ( x + y ) ) - b x y + c y dy/dt = bxy - cy

It is assumed that(i)the susceptible population grows until the total population (x+ y) reaches1(= the carrying capacity of the environment), (ii)the infection of the disease changes susceptible individuals into infected ones, and (iii)the infected individuals recover and become susceptible again at a certain rate.

Answer the following.

1.Explain how each of the above three assumptions were represented in the equations. (5x3 = 15points)

2.Find the equilibrium points. There are three such points. (5x3 = 15points)

3.Calculate the Jacobian matrix of the model. Keep a, b and c as symbols. (10 points)

4.Conduct linear stability analysis for each equilibrium point and discuss the conditions under which each equilibrium point is stable/unstable. (5x3 = 15points)

5.Identify the critical condition of parameter values at which a bifurcation occurs.Note that all the parameters are positive.(5points)

6.Draw the phase spaces of this model using Python for several different parameter values to confirm the prediction of bifurcation derived in the previous question.(10points)