Find the periodic payment R required to amortize a loan of P dollars over t years with interest charged at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.)

a. P = 50,000, r = 4, t = 15, m = 4

b. P = 90,000, r = 3.5, t = 17, m = 12'

c. P = 120,000, r = 5.5, t = 29, m = 4

A cup of hot coffee has a temperature of 201˚F when freshly poured, and is left in a room at 72˚F. One minute later the coffee has cooled to 191˚F.

a. Assume that Newton's Law of cooling applies. Write down an initial value problem that models the temperature of the coffee.

u'= -k(_____)

u(o)= _____

b. Determine when the coffee reaches a temperature of 183˚F. Give answer in minutes, and round the answer to two decimal places.

Solve the following linear equations with constant coefficients, using characteristic equations and undetermined coefficients as needed.

y''+4y'-12y=x+e^2x y(0)=1,y'(0)=2

Use the truth-tree decision procedure (relying on Proof Tools or pen/pencil and paper) to determine whether P→Q,Q⊨P is deductively valid.

A manufacture of laptop computers has four models, two with color lcd (liquid crystal display) screens (the spot model and the superba model) and two model with black and white lcd screens ( the standard model and the excel model). each model required assembly and test time and the requirements are shown in the table 2, together with the amount of time available for assembly and testing next month. LCD.

the lcd screens are purchased from and outside supplier and, because of an earthquake on japan (where the lcd screens are produced), they are in short supply. the outside supplier indicates that not more than 900 LCD screens in total could be supplied in the next month, and of these, not more than 500 could be color LCD.

it is possible for the manufacturer to make available additional hours of test time by adding a third shift at the manufacturing facility and paying overtime wages. up to 500 additional hours are available using this approach, but the premium cost is $10 per hour over that for regular time. the manufacturer can sell all the laptops of any model that can be produced. assume that the manufacturer is interested in maximizing the contribution from laptop computers (less any overtime . formulate this problem as a linear programming problem.

s standard model    excel model    sport model superba model    total available

assembly time    8    10 12 15 10,000

test time(HRS) 2    3 4    6 2,500

profit contribution    $120    $180    $240    $300

Find the solution of the initial value problem y′′+12y′+32y=0, y(0)=12 and y′(0)=−84.

FIND THE CENTROID:

a. of the region enclosed between the curves y=x^1/2 , y=1, y=2 and the y-axis.

b. of the 1st quadrant area bounded by the curve y=4-x²

c. of the region bounded by the curve y=x³ and x=y²

To appreciate how viscous drag are being computed if we have the velocity field, let us consider the following velocity field for a flow of Newtonian viscous fluid u = x + 2y v = x 2 + y w = 0 (4.158) The fluid has viscosity of 10−2 kgm−1 s −1 (a) Determine the viscous stresses on the reference fluid element.(b) Determine the viscous stress vector acting on a surface aligned at an angle of 45 degrees (counterclockwise) from the x−axis.(c) Determine the viscous force acting on the same surface if surface spans from x = 2 to x = 5 and its length in the z-direction being 2 m.

Write the proof of the dual Pappus theorem.

Find the general solution of the differential equation y′′+36y=13sec^2(6t), 0<t<π/12.

Consider these 2 functions (all with domain and codomain (Z/pZ) for some big prime p): h(x) = 1492831*x and h(x) = x³ . Why are these bad cryptographic hash functions? Give different reasons for the two.