Solve using Laplace Transform:

1) y'' - 2y' + 5y = cos(2t) - cos(2t)u_4pi(t); y(0) = 0, y'(0) = 0

Universal Mines Inc. operates three mines in West Virginia. The ore from each mine is separated into two grades before it is shipped. The daily production capacities of the three mines, as well as their daily operating costs, are as follows:

Universal has committed itself to deliver 54 tons of high-grade order and 65 tons of low-grade ore by the end of the week. Universal can run its mines seven days a week if required.

Determine the number of days each mine should be operated during the upcoming week if Universal Mines is to fulfill its commitment at the minimum total cost. Round the answers to two decimal places.

a. The number of days Mine I should operate =                            [ Select ]                       ["2.65", "5.94", "1.75", "1.00"]         days

b. The number of days Mine II should operate =                            [ Select ]                       ["3.95", "7.00", "4.06", "5.23"]         days

c. The number of days Mine III should operate =                            [ Select ]                       ["6.00", "7.00", "2.95", "5.00"]         days

d. The total cost of the operation for next week = $                            [ Select ]                       ["279,000", "284,000", "320,000", "245,000"]      

How many equations would you need to compute a series of cubic splines of the form ax^3 + bx^2 + cx + d to fit through 4 data points and why?

Determine the solution of a homogeneous linear first order Ordinary Differential Equation system: (use 2 methods, substitution method, and matrix method)

1. x1' = - 4x₁-6x₂

           x2' = x₁+ x₂

With the initial values: x₁ (0) = 2 ; x₂ (0) = -1

b.  x₁' = -x₂

             x₂' = -x₁

With initial values: x₁ (0) = 3 ; x₂ (0) = 1

You have a jar containing pennies, nickels, dimes, and quarters. In how many ways can you select exactly 10 coins if the order in which you select them does not matter and

1. You have at least 10 of each type of coin available and there are no restrictions

2. You have at least 10 of each type of coin available and you need to select at least two dimes and two nickels

3. You have at least 10 pennies, 10 nickels, and 10 dimes available, but only 6 quarters

Let D3 be the symmetry group of an equilateral triangle. Show that the subgroup H ⊂ D3 consisting of those symmetries which are rotations is a normal subgroup.

Prove that if a sequence is bounded, then limsup sn is a real number.

Find the particular solution:

y″−4y′+4y=(−11.5e^(2t))/(t^(2)+1)

Show that a point x is an accumulation point of a set E if and only if for every ε > 0 there are at least two points belonging to the set E ∩ (x − ε, x + ε).

A binary string is a “word” in which each “letter” can only be 0 or 1

Prove that there are 2^n different binary strings of length n.

Note:

• Your goal is to produce a properly constructed proof by induction, but this does not mean you have to follow
• Mathematical induction, step-by-step..
• Write the statement with n replaced by k
• Write the statement with n replaced by k+1.
• Identify the connection between the kth statement and the (k+1)th statement.
• Complete the induction step by assuming that the n=k version of the statement is true, and using this assumption to prove that the n=k+1 version of the statement is true.
• Complete the induction proof by proving the base case.
• point-by-point.
• ANOTHER IMPORTANT NOTE:
• Make sure you are addressing the given statement: "there are 2ⁿ different binary strings of length n” !!! So your solution should primarily be discussing binary strings. Yes, the fact that 2^k+1 = 2^k∙2 will be useful in your solution, but it should not be the sole content of your solution, as by itself that equality is a fact about exponents, not a fact about binary strings.

Helen purchased a house for $450,000. She made a downpayment of 15% of the value of the house and received a mortgage for the rest of the amount at 5.50% compounded semi-annually for 15 years. The interest rate was fixed for a 5-year term.

a. Calculate the size of the monthly payments.

Round to the nearest cent

b. Calculate the principal balance at the end of the 5-year term.

Round to the nearest cent

c. Calculate the size of the monthly payments if after the first 5-year term the mortgage was renewed for another 5-year term at 5.25% compounded semi-annually?

Round to the nearest cent

use bisection to find the real root x = sin(x) + 1 with the initial guesses of x l = 0 and x u = 3.

perform the computation until the approximate error falls below 5%

Let a be a positive element in an ordered field. Show that if n is an odd number, a has at most one nth root; if n is an even number, a has at most two nth roots.

Hampton Commission on the Arts (HCA) is planning on advertising its upcoming events. A local newspaper claims that the ads on the paper can reach 27,000 art lovers, and it costs the HCA $6,000. Advertising on a local radio $10,000, and it is estimated to reach an audience of 39,000 people. HCA also considers google ads, which costs $15,000 will supposedly reach 52,000 arts customers. The breakdown of the audience for each type of commercial is as follows (charts below):

HCA has determined some marketing guidelines in order to improve outreach efforts. It wants to reach at least 200,000 potential arts customers. Past experience shows that older people are more likely to buy tickets than younger people; the number of potential customers over 35 should be at least 50% more than the potential customers under 35. The council also believes women are more likely to initiate ticket purchases to the arts than men, so it wants its advertising audience to be at most 40% male.

1) Construct a mathematical model for HCA to determine the number of ads of each type it should use at the minimum cost.

2) Solve the model using Excel Solver