solve this IVP

9y'' +33.33y' +464.21y=8sin(t/4), y(0)=0, y'(0)=.04

Suppose A = {(a, b)| a, b ∈ Z} = Z × Z. Let R be the relation define on A where (a, b)R(c, d) means that 2 a + d = b + 2 c.

a. Prove that R is an equivalence relation.

b. Find the equivalence classes [(−1, 1)] and [(−4, −2)].

Find the Fourier series expansion of the function f(t) = t + 4 , 0 =< t < 2pi

A mass weighing 17 lb stretches a spring 7 in. The mass is attached to a viscous damper with damping constant 2 lb *s/ft. The mass is pushed upward, contracting the spring a distance of 2 in, and then set into motion with a downward velocity of 2 in/s. Determine the position u of the mass at any time t. Use 32 ft/s^2 as the acceleration due to gravity. Pay close attention to the units. Leave answer in terms of exact numbers(no decimals).

A large tank containing a mystery liquid is filled to a depth of L = 35 m. The upper surface of the liquid is exposed to the atmosphere (of density 1.2 kg/m³). A pipe of cross-sectional area A_in = 0.01 m² is inserted in to the liquid. The other ’outlet’ end of the pipe, of smaller cross sectional area A_out = 0.005 m², is placed outside the liquid at a height of h = 2 m below the surface of the liquid. Fluid begins to flow out of the outlet.

a (7 points) Someone submerges an object of density 900 kg/m³ in the mystery liquid, and it floats suspended (a = 0). What is the density of the mystery liquid?

b (7 points) Find the speed v_out of the liquid flowing out of the outlet.

c (6 points) Find the speed v_in of the liquid flowing into the pipe.

d(5points) Now a fierce wind with v_wind =35 m/s blows parallel to the entire surface of the tank exposed to the air (it does not reach or affect the air around the outlet). This lowers the pressure on the liquid at the top of the tank (but liquid does not spill over the top). With the wind blowing, how much slower is the liquid flowing out of the outlet?

Write a recursive formula that shows how many ways you can tile a 3xn checkboard with 1x3 tiles. Show how the pattern is establish by showing how different value n give their corresponding ways to tile the particular n.

Demand for propane is given by D(x) = 6.5 − 0.25x, and supply is given by S(x) = 2.1 + 0.15x, where x is in gallons per month customer and D(x) and S(x) are dollars per gallon. Find the followings:  

(a) Equilibrium point (X_e, P_e)  

(b) Find the consumer surplus and the producer surplus at the equilibrium point.

(c) Assume a price ceiling of $3 per gallon of propane is imposed. Find the point (X_c, P_c)  

(d) Find the new producer surplus and the new consumer surplus at (X_c, P_c).

(e) Find the deadweight loss.

5.       Solve the following differential equation using the given initial conditions (Use convolution and set up the integral but do not integrate.)

              y^'’ − 2y^’ + 2y = 18e^−t sin3t;         y(0) = 0, y^’(0) = 3

Show all work, write legibly, explain in detail

y"-3y'+2y = 8u₂(t) , y(0) = 0, y'(0) = 0

Solve the following initial value problem using the Laplace transform

Question Set 2: Two Independent Means

Answer the following questions using the NYC2br.MTW file. You can find this dataset in this assignment in Canvas (i.e., where you downloaded this document and where you’ll upload your completed lab). Data were collected from a random sample of two-bedroom apartments posted on Apartments.com in Manhattan and Brooklyn.

A. What is one type of graph that could be used to compare the monthly rental rates of these two-bedroom apartments in Manhattan and Brooklyn? Explain why this is an appropriate graph. [10 points]

B. Using Minitab Express, Construct the graph you described in part A to compare the Manhattan and Brooklyn apartments in this sample. [10 points]

C.  Use the five-step hypothesis testing procedure given below to determine if the mean monthly rental rates are different in the populations of all Manhattan and Brooklyn two-bedroom apartments. If assumptions are met, use a t distribution to approximate the sampling distribution. You should not need to do any hand calculations. Use Minitab Express and remember to include all relevant output. [30 points]

please use minitab. thanks!

Purchase of a new home

John and Jane had planned to save $60 000 dollars over the next five years as a down payment on a house. Jane assured John that if they contributed $1000 each month to a savings account that pays an annual rate of interest of 2.5% compounded monthly that they would have enough money to put a down payment of $60 000 on their new house. Wanting their daughter to have a house, Jane’s parents (The Henrys) have offered to lend John and Jane $65 000, which they have suggested (perhaps naively) John and Jane pay back by contributing to a savings account in the Henrys’ name as per Jane’s original savings plan. John’s worried this is not fair to his in-laws. Is he correct? If so, devise a fair repayment plan that would see the Henrys repaid at a rate of 2.5% compounded monthly over the 5 years.

The Does have qualified for a mortgage of $500,000 to be amortized over 25 years. Their mortgage broker has offered them the following options:

1. A 5 year fixed rate with monthly payments at an annual interest rate of prime+1%
2. A 10 year fixed rate with biweekly payments at an annual interest rate of prime+2%

Prime is currently at 1.5% and projected to increase by 0.25% every year for the next 10 years. Which Mortgage terms should they accept given that their goal is to pay as much principle as possible over the next 10 years?

Teacher's notes:

To begin you know the payment size, the number of payments and the interest rated and what you need to determine is the FV of those payments using the formula: FV=PMT[((1+i)^n - 1)/i].  

This will tell you how much money the Does will save (or pay back to the Henrys) under Janes original plan. You can use this same formula to ask how big the PMTs would have to be to ensure the Does pay back the Henrys exactly 65 000 dollars. Of course this means the Henrys earn no interest.

If you want to ensure the Henrys earn interest as per Jane’s original payment plan, then you’ll need to calculate the PMTs needed based on a present value of 65000 using the formula: PV=PMT((1-(1+i)^-n)/i). This will tell you how big the PMTs would have to be to ensure the Does pay back the Henrys 65 000 dollars with interest as per Jane’s original plan.

It is like a class discussion and we're supposed to write a discussion about Vector spaces, subspaces and bases

Overview (what to write on the discussion): So we're supposed to discuss and explain about these following points:

1. Vector Spaces, Definitions, Examples, Non Examples
2. Spanning Sets, Linear Independence
3. Basis, Dimension
4. Rank
5. Change of Basis, Coordinates

PUT EXAMPLES AND DEFINITIONS (EX: Showing examples to vector addition (closure under addition, etc) and scalar multiplication (distributive property, etc)

Instructions:

Add a new discussion topic. In your post, please include the following:

1. Details on proofs.
2. Illustrate examples of special interest.
3. Apply your mathematical knowledge of given structures in proving or disproving assertions regarding Vector Spaces.

The Tax Cuts & Jobs Act enhanced the deduction for charitable contributions by raising the limit that can be contributed in any one year. The limit is now 60% of adjusted gross income (AGI), up from 50%. Assume your client still has a charitable deduction limitation due to their AGI. The client might lose the charitable deduction because of achieving this limitation.

• Recommend at least two tax planning strategies to avoid losing the deductions. Provide support for your response

Solve by variation of parameters.

y''+4y =sin(2x)

y'''-16y' = 2

Let S be a set of n numbers. Let X be the set of all subsets of S of size k, and let Y be the set of all ordered k-tuples

(s₁, s₂,   , s_k)

such that

s₁ < s₂ <    < s_k.

That is,

(a) Define a one-to-one correspondence

f : X → Y.

Explain why f is one-to-one and onto.

(b) Determine |X| and |Y|.