This problem deals with the RC circuit shown to the right containing a resistor​ (R ohms), a capacitor​ (C farads), a​ switch, a source of​ emf, but no inductor. The differential equation for this circuit is given below.

R(dQ/dT)+1/c(Q)=E(t)

this is an equation for the charge Q(t) on the capacitor at time t. Note that I(t)=Q'(t). Use the values R=100, C=5.0x10^-4, Q(0)=0, and E(t)=100cos120t to find (a) and (b) below:

a: find Q(t) and I(t)

b: What is the amplitude of the​ steady-state current?

Complex Variable Evaluate the following integrals:

a) int_c (z^2/((z-3i)^2)) dz; c=lzl=5

b) int_c (1/((z^3)(z-4))) dz ; c= lzl =1

c) int_c (2(z^2)-z+1)/(((z-1)^2)(z+1)) dz ; c= lzl=3

(Details Please)

Consider a projectile launched at a height h feet above the ground and at an angle θ with the horizontal. If the initial velocity is v₀ feet per second, the path of the projectile is modeled by the parametric equations

x = t(v₀ cos(θ))

and

y = h + (v₀ sin(θ))t − 16t².

The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit h = 3 feet above the ground. It leaves the bat at an angle of θ degrees with the horizontal at a speed of 107 miles per hour (see figure).

(a) Write a set of parametric equations for the path of the ball. (Write your equations in terms of t and θ.)

(b) Use a graphing utility to graph the path of the ball when θ = 15°. Is the hit a home run?

YesNo    

(c) Use a graphing utility to graph the path of the ball when θ = 23°. Is the hit a home run?

YesNo    

(d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run. (Round your answer to one decimal place.)
  °

Find the smallest positive integer x that satisfies the system of congruences

x ≡ 3 (mod 5).

x ≡ 5 (mod 7).

x ≡ 7 (mod 11)

Find the system of equations to model the problem. Don't solve the system.

A paper company produces high grade, medium grade and low grade paper. The number of tons of each grade produced from a ton of pulp depends on the source of the pulp. The following table shows three different sources and the amount of each grade of paper that can be produced from one tonne of pulp from each source.

(Number of tons)

High grade Medium grade Low grade
Pulp from Brazil 0.6 0.3 0.1
Domestic pulp 0.5 0.3 0.2
Recycled pulp 0.3 0.4 0.3

The company has received an order for 11 tons of high grade paper, 15 tons of medium grade paper, and 14 tons of low grade paper. How many tons of each type of pulp must be used to accurately fulfill this order? Establish a system of linear equations. Let x, y, and z equal the number of tons of Brazilian pulp, domestic pulp, and recycled pulp, respectively, needed to fulfill the order.

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!