In 6 years Harry and Sally would like to have $28,000.00 for a down payment on a house. How much should they deposit each month into an account paying 5% compounded monthly?

5.State and proof the existence and uniqueness of the Laplace transform.

6.State and proof the linearity properties of the Laplace transform.

18.State and proof the first shifting theorem.

Consider functions f : {1,2,3,4}→{1,2,3,4,5,6}.

a) Give an example of one such function (use 2-line notation). Then say how many such functions there are and why your answer makes sense.(25 points)

(b)Give one example of such a function that is injective and one that is not. Then say how many injective functions there are and why your answer makes sense.(50 points)

(c) Explain why there are no surjective functions with this domain and codomain. (25 points)

Let a_n denote the number of sequences of 0's and 1's that do not contain two consecutive 0's. Determine a_n.

Course: Discrete Math

Show/Prove that the principal of strong induction and regular induction are equivalent.

(Transient Current)
An RLC series circuit has a voltage source given by E(t) = 20sin(t) V, an inductor of 3 H, a resistor 6 Ω, and a capacitor of 1/3 F. Find the current I(t) in this circuit for t > 0 if I(0) = 2, I'(0) = 0. Find the moment of time when the transient current is equal to zero

Portsmouth Company makes upholstered furniture. Its only variable cost is direct materials. The demand for the company's products far exceeds its manufacturing capacity. The bottleneck (or constriant) in the production process is upholstery labor-hours. Information concerning three of Portsmouth's upholstered chairs appears below:

Required:

1. Portsmouth is considering paying its upholstery laborers additional compensation to work overtime. Assuming that this extra time would be used to produce sofas, up to how much of an overtime premium per hour should the company be willing to pay to keep the upholstery shop open after normal working hours?

2. A small nearby upholstering company has offered to upholster furniture for Portsmouth at a price of $50 per hour. The management of Portsmouth is confident that this upholstering company’s work is high quality and their craftsmen can work as quickly as Portsmouth’s own craftsmen on the simpler upholstering jobs such as the Love Seat. How much additional contribution margin per hour can Portsmouth earn if if it hires the nearby upholstering company to make Love Seats?

3. Should Portsmouth hire the nearby upholstering company?

A small nearby upholstering company has offered to upholster furniture for Portsmouth at a price of $50 per hour. The management of Portsmouth is confident that this upholstering company’s work is high quality and their craftsmen can work as quickly as Portsmouth’s own craftsmen on the simpler upholstering jobs such as the Love Seat. How much additional contribution margin per hour can Portsmouth earn if if it hires the nearby upholstering company to make Love Seats?

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6. Consider the one dimensional wave equation with boundary conditions and initial conditions:

PDE : utt = c 2 uxx, BC : u(0, t) = u(L, t) = 0, IC : u(x, 0) = f(x), ut(x, 0) = g(x)

a) Suppose c = 1, L = 1, f(x) = 180x 2 (1 − x), and g(x) = 0. Using the first 10 terms in the series, plot the solution surface and enough time snapshots to display the dynamics of the solution

b) What happens to the solution as t → ∞? Explain your answer in light of (a) and the physical interpretation of the problem. Does (b) reflect this?

c) Redo parts (a) and (b) for c = 1, L = π, f(x) = 2 sin 3x, and g(x) = 1 − x.

The Poisson probability distribution, associated with a waiting line, is characterized by a parameter known as arrival rate. Explain this concept and describe its connections with the management of a waiting line.

Boundary-Value Problems in Other Coordinate Systems

1. Solve ∆u = 0 in a disk x^2 + y^2 ≤ 25, where u(5, θ) = 7 sin 3θ − 6 sin 8θ and u is bounded when   r = 0.
2. Solve ∆u = 0 in an annulus 1 ≤ x^2+y^2 ≤ 4, where u(1, θ) = 75 sin θ, u(2, θ) = 60 cos θ.
3. Find the steady-state temperature distribution in a disk of radius 1 if the upper half of the circumference is kept at 100◦ and the lower half is kept at 0◦ .

Find Mean, median, varianca and standard deviation for:
6, 10, 18, 25, 13, 16, 11, 7, 11.

Jones Company provided the following information:

Fair value of the reporting unit, including goodwill $1,400,000

Book value of reporting unit, excluding goodwill $1,000,000

Add: Carrying value of goodwill 600,000

Carrying value of the reporting unit, including goodwill $1,600,000

The qualitative assessment of goodwill is completed and it is more likely than not that goodwill is impaired. Describe the process for determining if Jones needs to record a goodwill impairment loss and prepare any required journal entries.

Two tanks contain a mixture of water and alcohol with tank A contain- ing 500 L and tank B 1000L. Initially, the concentration of alcohol in Tank A is 0% and that of tank B is 80%. Solution leaves tank A into B at a rate of 15 liter/min and the solution in tank B returns to A at a rate of 5 L/min while well mixed solution also leaves the system at 10 liter/min through an outlet. A mixture of water and alcohol enters tank A at the rate of 10liter/min with the concentration of 10% through an inlet. What will be the concentration of the alcohol of the solution in each tank after 10 mins?