Problem #1 :

Using separation of variables and the formalism demonstrated in class, find the general solution to the Helmholtz equation (also known as the time independent wave equation) :

∇^2 F + k^2 F = 0

Assume that k is a real number. For clarification, the general solution is the solution in the case where boundary conditions are not specified.

Problem #2 :

Express the following functions as an infinite Fourier sine series using sin(nπx/a)

a)f(x) = x

b)f(x) = e^x

c)f(x) = cos(x)

Quiz 4

A manufacturer makes and sales four types of products:  Product X, Product Y, Product Z, and Product W.  

The resources needed to produce one unit of each product and the sales prices are given in the following Table.

• Currently, 4,600 pounds of steel and 5,000 machine hours are available.

• To meet customer demands, exactly 950 total products must be produced.

• Customers also demand that at least 400 units of Product W be produced.

Formulate an LP that can be used to maximize sales revenue for the manufacturer.

LP Formula

Let Pi be the number of product type i produced by the manufacturer, where i = X, Y, X, and W.

MAXIMIZE  4 PX + 6 PY + 7 PZ + 8 PW

Subject To

2 PX + 3 PY + 4 PZ + 7 PW <= 4600   ! Available Steel

3 PX + 4 PY + 5 PZ + 6 PW <= 5000   ! Available Machine Hours

PX + PY + PZ + PW    = 950               ! Total Demand

                            PW >= 400               ! Product W Demand

PX >=0

PY >=0

PZ >=0

PW >=0

Suppose manufacturer raises the price of Product Y by 50¢ per unit. What is the new optimal solution to the LP?

1. How is the ratio of outputs to inputs, y x, different from the slope ratio?
2. When a linear graph passes through the origin, why is the ratio of outputs to inputs, y x the same as the slope?
3. How can you tell from a graph whether a relationship is inverse variation or direct variation? Does this change when we look at direct square variation versus inverse square variation? Why or why not?
4. What is a horizontal asymptote? Do direct or inverse variation graphs have asymptotes? Why?
5. Explain how to tell whether a table varies directly, inversely, or neither. Include what to watch for to determine if a function varies directly or inversely with the square.

** NEED MATLAB**

design a cam with harmonic and cycloidal rise and 3-4-5 polynomial fall

specifications:

*Cycloidal rise (0◦ < θ < 80◦) from 0 mm to 20 mm

*Dwell (80◦ < θ < 100◦)

*Harmonic rise (100◦ < θ < 180◦) from 20 mm to 30 mm

*Dwell (180◦ < θ < 210◦)

*3-4-5 Polynomial fall (210◦ < θ < 300◦) from 30 mm to 0 mm

*Dwell (300◦ < θ < 360◦)

•the radius of the base circle is 40 mm, radius of follower is 5 mm, and cam is driven by a constant speed motor rotating counter-clockwise at 500 rpm with the above specifications

----------------------

1. Plot the displacement, velocity, accleration, and jerk profiles for one revolution

2. Plot the pressure angle as a function of θ (eccentricity is 0)

3. Plot cam contour, pitch curve, prime circle, and base circle (all on same plot)

4. Plot the prime circe and cam contour at various orientations (from θ =0◦ to θ =240◦. AND arrange 9 subfigures as a 3×3 matrix)

How could a function be defined everywhere but continuous at only one point?

The population of mosquitoes in a certain area increases at a rate proportional to the current population, and in the absence of other factors, the population doubles each week. There are 800,000 mosquitoes in the area initially, and predators (birds, bats, and so forth) eat 60,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time. (Note that the variable t represents days.)  

Determine the entropy of the random variable which counts the sum of
three dice.

How to determine whether the following statements about big-O notation are true or false?

(a) Let f(n) = √ n log n − 4, then f(n) = O(n^ 2)

(b) Let f(n) = 4 n + 2 log^ 2 (n), then f(n) = O(log^ 2 (n))

(c) Let f(n) = 5 √ n + 2, then f(n) = Ω(log^ 4 (n))

(d) Let f(n) = 5 n^ 2 + 5 n log n + 4, then f(n) = O(n^3 )

(e) Let f(n) = 2 log^2 (n) + 5√ n + 7, then f(n) = Θ(√ n)

The number of hours worked by 24 employees of a company is given below:
40 43 40 39 36 44 40 39 39 52 27 50
41 47 40 48 38 36 25 41 35 36 16 40
(a) (6 points) Calculate the mean, variance and standard derivation for the given data
(b) (6 points) Calculate the three quartiles (Q1, Q2, and Q3) and the Interquartile range
(IQR)
(c) (4 points) Calculate the values of the lower fence and the upper fence for a boxplot.
(d) (5 points) Construct a box-and-whisker plot. Comment on the shape of the distri-
bution of the data. List any potential outliers, if any

2) A drug lord wishes to optimize the area in which he will grow his coca plants. The area concerned has a river on one side and must be closed in with fencing on the other three sides. He has 500ft of fencing. Determine the dimensions of the field that will maximize the area. Show that this is indeed a maximum.

3) The drug lord is now in jail. He requests that there be a window with a semicircular top in his cell that lets in the most amount of light possible. If there are 20m of framing material available, what are the dimensions of the window? Again, show that this is indeed a maximum.

Quiz 4

A manufacturer makes and sales four types of products:  Product X, Product Y, Product Z, and Product W.  

The resources needed to produce one unit of each product and the sales prices are given in the following Table.

• Currently, 4,600 pounds of steel and 5,000 machine hours are available.

• To meet customer demands, exactly 950 total products must be produced.

• Customers also demand that at least 400 units of Product W be produced.

Formulate an LP that can be used to maximize sales revenue for the manufacturer.

LP Formula

Let Pi be the number of product type i produced by the manufacturer, where i = X, Y, X, and W.

MAXIMIZE  4 PX + 6 PY + 7 PZ + 8 PW

Subject To

2 PX + 3 PY + 4 PZ + 7 PW <= 4600   ! Available Steel

3 PX + 4 PY + 5 PZ + 6 PW <= 5000   ! Available Machine Hours

PX + PY + PZ + PW    = 950               ! Total Demand

                            PW >= 400               ! Product W Demand

PX >=0

PY >=0

PZ >=0

PW >=0

Suppose the sales price of Product Z is decreased by 60¢. What is the new optimal solution to the LP?

Quiz 4

A manufacturer makes and sales four types of products:  Product X, Product Y, Product Z, and Product W.  

The resources needed to produce one unit of each product and the sales prices are given in the following Table.

• Currently, 4,600 pounds of steel and 5,000 machine hours are available.

• To meet customer demands, exactly 950 total products must be produced.

• Customers also demand that at least 400 units of Product W be produced.

Formulate an LP that can be used to maximize sales revenue for the manufacturer.

LP Formula

Let Pi be the number of product type i produced by the manufacturer, where i = X, Y, X, and W.

MAXIMIZE  4 PX + 6 PY + 7 PZ + 8 PW

Subject To

2 PX + 3 PY + 4 PZ + 7 PW <= 4600   ! Available Steel

3 PX + 4 PY + 5 PZ + 6 PW <= 5000   ! Available Machine Hours

PX + PY + PZ + PW    = 950               ! Total Demand

                            PW >= 400               ! Product W Demand

PX >=0

PY >=0

PZ >=0

PW >=0

Suppose that 4,500 pounds of steel are available. What is the new optimal z-value?

What if only 4,400 pounds of steel are available? What is the new optimal z-value?

What is the most that manufacturer should be willing to pay for an additional unit of raw material?

What is the most that manufacturer should be willing to pay for an additional unit of machine hour?

In order to answer the above questions, how many times you did update your model and solve it?

Prove that √3 is irrational using contradiction. You can use problem 4 as a lemma for this.

Problem 4, for context is

Prove that if n² is divisible by 3, then n is divisible by 3.

A list of six positive integers, p, q, r, s, t, u satisfies p < q < r < s < t < u. There are exactly 15 pairs of numbers that can be formed by choosing two different numbers from this list. The sums of these 15 pairs of numbers are: 25, 30, 38, 41, 49, 52, 54, 63, 68, 76, 79, 90, 95, 103, 117. Which sum equals r + s?