A cylindrical tower has radius R meters and a roof which is a hemisphere. The total height of the tower is R + H meters.

a)Sketch the tower and label its various measurements using cylindrical coordinates. Be sure to include a function z = f(r, θ) whose graph is the top of the tower.

b)Write an integral in cylindrical coordinates to express the volume of the tower.

c)Verify your integral gives the right volume, according to the formula πR^2H for the volume of a cylinder and 4/3πR^3 for the volume of a ball.

Please explain in detail and correctly. Thanks

The relations: P = {(x,y) ∈ R×R: x = y²+2} and Q = {(x,y)  ∈ R×R : x = 2y}

a) P^-1

^b) P ◦ Q

c)  Rng(P^-1 ◦ Q^-1)

Find the optimum solution to the following LP by using the Simplex Algorithm.

Min z = 3x1 – 2x2+ 3x3

s.t.
-x1 + 3x2 ≤ 3

x1 + 2x2 ≤ 6

x1, x2, x3≥ 0

a) Convert the LP into a maximization problem in standard form.

b) Construct the initial tableau and find a bfs.

c) Apply the Simplex Algorithm.

Prove:

Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

Let x, y ∈ Z. Prove that x ≡ y + 1 (mod 2) if and only if x ≡ y + 1 (mod 4) or x ≡ y + 3 (mod 4)

Solve the differential equation using the method of undetermined coefficients.

y'' − 10y' + 26y = e^−x

Elementary Differential Equations Problems:

1) Find the solution of the initial value problem of y" + 3y' = 0, y(0) = -2, y'(0) = 3

2) Find the general solution of the equation 4y" - 9y = 0

3) Find the general solution of the equation dy/dt = 2t(y – 2y²)

4) Given the second order linear homogeneous equation y"- 2y' + y = 0,

a) Verify that y₁(t) = e^t and y₂(t) = t e^t are solutions of the equation

b) Compute the Wronskian of y₁, y₂

c) Would y(t) = C₁y₁(t) + C₂y₂(t) also be a solution to the equation for arbitrary constants C₁,C₂? Explain.

5) Consider the second order non-linear differential equation (y * y") + (y’)² = 0 for t > 0

a) Verify that y₁(t) = 1 and y₂(t) = t^1/2 are solutions of this equation

b) Let y(t) = C₁y₁(t) + C₂y₂(t) = C₁+ (C₂ * t^1/2), for constants C₁,C₂. Compute y' and y"

c) Use the results from above to show that y(t) is NOT always a solution of the equation (y * y") + (y’)² = 0 for all C₁, C₂

If graph G has n edges and k component and m vertices, so m ≥ n-k. Prove it!

Classify the following equation and reduce it to canonical form:

yUxx-xUyy=0 , x>0 , y>0

With the completion of the determinations of % potassium, % iron, and % oxalate in the crystals, you may calculate the % water. The percentage compositionof the crystals, K_xFe(C₂O₄)_y · zH₂O, has then been completely determined experimentally. The simplest formula (x,y,z) can now be calculated from the the percentage composition. Once the formula is know it is then possible to calculate the percent yield of product that was obtained in the preparation and purification of the crystals.

From Part A:

Mass of K_xFe(C₂O₄)_y · zH₂O prepared : 4.800 g
Mass of FeCl₃ : 1.60 g

From Part B:

% Potassium in compound : 18.97 %
% Iron (from ion exchange & titration vs. NaOH) : 10.95 %

From Part C:

% Oxlate : 43.50 %

The questions below are part of the final analysis

Calculate the % water of hydration : 26.58

Calculate the following for Fe³⁺:

Calculate the following for K⁺:

Calculate the following for C₂O₄^2-:

Calculate the following for H₂O

Enter the simplest formula of the Iron Oxalate Complex Salt:

Now that the formula of the complex salt is known, the percent yield can be determined.

Calculate the moles of FeCl₃ used in preparation:

Calculate the theoretical moles of K_xFe(C₂O₄)_y · zH₂O:

Calculate the actual moles of K_xFe(C₂O₄)_y · zH₂O synthesized:

Calculate the percent yield:

Let A be a set of real numbers. We say that A is an open set if for every x₀ ∈ A there is some δ > 0 (which might depend on x₀) such that (x₀ − δ, x₀ + δ) ⊆ A. Show that a set B of real numbers is closed if and only if B is the complement of some open set A

Diff. equations

Consider the equation (x^2 − 2)y''+ 3xy'+ y = 0.

a) Find the general solution as a power series centered at x = 0. Write the first six nonzero terms of the solution. And write the solution using sigma notation with a formula for the coefficients. Write the two linearly independent solutions that form the general solution.

b) Find a power series solution satisfying the initial conditions y(0) = 2 and y' (0) = 3. Write the first six nonzero terms of the solution. And write the solution using sigma notation with a formula for the coefficients.

You purchase a house that costs 800000TL. For this you take out a loan from a bank for 25 years at 8.8% interest. You will pay your debt by making equal monthly payments. After 15 years the interest rates have dropped. Another bank offers you 7.5% interest. You decide to take out a second loan for the next 10 years, which will be paid monthly to this new bank and you pay off your total debt to the first bank.

• How much were your monthly payments for the first 15 years?

• How much interest is paid between years 12 and 15?

• How much are your new monthly payments for the remaining 10 years?

• How much money is saved in 25 years by doing this refinancing?

Given the function f(x) on the right solve the following root finding questions: a) Find a positive root (x > 0) of f(x) using the Bisection Method

. b) Find a negative root (x < 0) of f(x) using the Bisection Method.

c) Find a positive root (x > 0) of f(x) using the False Position Method.

d) Find a negative root (x < 0) of f(x) using the False Position Method. Find your initial Bracket via Trial-and-Error. Use | fM | < 0.0001 as the Stopping Criteria. Calculate and present all quantities with at least 4 decimal digits. Examine function f(x) whose plot given below. Major grid lines correspond to whole units and minor grid lines correspond to quarter units. Let f'(x) be the derivative of f(x) and F(x) be the indefinite integral of f(x).

f(x) = 0.5*x² + 10*x - 50 / x² - x + 10