You have entered a model rocket contest with your friend, Tiffany. You have been working on a pressurized rocket filled with nitrous oxide. Tiffany has determined the minimum atmospheric pressure at which the rocket fuel is stable. Based not hat value, and the equations given below, your task is to determine the optimum launch angle and initial velocity to maximize flight time. The goal is to re-use your rocket capsule, so you really want to avoid a fuel explosion.

The atmospheric pressure varies with elevation according to the equation: P(h)= 14.7e−h/10. where p is the pressure in psi and h, is an elevation in miles above sea levels. The height (in feet) of a rocket launched at an angle α degrees with the horizontal and an initial velocity, vo in feet/second, t seconds after launch is given by the equation h(t)=-16t^2+vo*t*sin(α).

1) If Tiffany has determined that the minimum safe pressure is 11 pounds per square inch, at what altitude will the rocket explode? Report your result in feet. Round to the nearest foot.

2) If the angle of launch is33o, with an initial velocity of 1,648 what is the minimum atmospheric pressure exerted on the rocket during its flight? Report your answer to one decimal place. Under these conditions, will the rocket explode during its flight?

3) If the angle of launch is32o, with an initial velocity of 1,908 what is the minimum atmospheric pressure exerted on the rocket during its flight? Report your answer to one decimal place. Under these conditions, will the rocket explode during its flight?

4) Tiffany has revisited her calculation and has now concluded that the minimum safe pressure for the fuel is 9 psi. What is the maximum height your rocket can achieve without exploding in flight? Report your answer in feet to the nearest foot.

5) Tiffany has (once again) checked her calculations, and you have verified with her that the safe pressure for your fuel is 9 and the fuel capsule holds enough fuel to produce an initial velocity of 2,169 feet per second. What launch angle will you use so that your rocket achieves the maximum safe altitude? Round your answer to the nearest tenth of a degree.

Euler’s Method Let’s get our hands dirty and actually use Euler’s method to estimate the value of y(2) where y is the solution to the initial value problem

y′=y−2x             y(0) = 1

Recall that Euler’s method says: Approximate values for the solution of the initial value problem

y′=F(x, y),y(x0) =y0 with step size h, at xn=xn−1+h, are

yn=yn−1+hF(xn−1, yn−1)

Fill in the table for steps of size h= 0.2.

Graph the portion of the approximate solution curve you found above. It should look like a lot of line segments. The first segment has been given on the grid below:

(c) Suppose f(x) is an exact solution to the initial value problem above. Describe, with justification, the behavior off(x) as x→∞. Hint: Graphing a slope field may be helpful for this.

Find a base of solutions. Try to identify the series as expansions of known functions. ( show details of your work)

xy''+2y'+xy=0 INTENTIONALLY +XY

1. Let g(s) = √ s. Find a simple function f so that f(g(s)) = 0. Hint: see Methods of computing square roots on Wikipedia. Use Newton’s method to estimate √ 2. Start with 3 different (and interesting) initial values of your choice and report the number of iterations it takes to obtain an accuracy of at least 4 digits.

In python.

y'' + 16y = (8)(cos(4t)) y(0)=y'(0)= 0

Use Laplace Transforms to solve. Sketch the solution or use matlab to show the graph.

Sec 5.4

-find the payment necessary to amortize a 12% loan of $2100 compounded quarterly with 15 quarterly payments

-find the monthly house payments necessary to amortize a 10.8% loan of 162,200 over 25 years

-find the payment made by ordinary annuity with the present value.

-77,822; monthly payments for 24 years; interest rate 4.3%, compounded monthly

-287,938; quarterly payments for 29 years; interest rate 6%, compounded quarterly

give an example of the divergence theorem and the greens theorem

Big Chill, Inc. sells portable dehumidifier units at the current price of $183. Unit variable costs are $75. Fixed costs, made up primarily of salaries, rent, insurance and advertising, are $4,018,000. Calculate breakeven sales for Big Chill, Inc. Round your answer to the nearest whole number.

A manufacturer is considering a switch from manufacturers’ representatives to an internal sales force. The following cost estimates are available. Manufacturers’ reps are paid 8.7% commission and incur $610,000 in fixed costs, while an internal sales force has fixed costs projected at $2,150,000 and would receive 2.7% commission. Assume that sales revenue is double the breakeven volume or the point at which the manufacturer would be indifference between reps and an internal sales force. At this volume, how much would the manufacturer save, assuming the company had switched to an internal sales force? Report your answer in dollars.

y'' - y = e^(-t) - (2)(t)(e^(-t)) y(0)= 1 y'(0)= 2

Use Laplace Transforms to solve. Sketch the solution or use matlab to show the graph.

y'' + 16y = (8)(cos(4t)) y0)= 0 y'(0)= 8

Use Laplace Transforms to solve. Sketch the solution or use matlab to show the graph.

1) Show the absolute value function f(x) = |x| is continuous at every point.

2) Suppose A and B are sets then define the cartesian product A * B

Please answer both the questions.

1.) (10pts) Consider the following differential equation: (x^2)(dy/dx)=2x(sqrt(y))+(x^3)(sqrt(y))

a)Determine whether the equation is separable (S), linear (L), autonomous (A), or non-linear (N). (An equation could be more than one of these types.)

b)Identify the region of the plane where the Chapter 1 Existence and Uniqueness Theorem guarantees a unique solution exists at an initial condition (x0, y0).

2.(12pts) Consider the IVP: y'+y=y/t , y(2) = 0

For each of the functions y₁(t)and y₂(t) below, decide if it is a solution of the IVP. (Answer is Yes or No, but show, or explain, briefly how you decided for each.)

(a) y₁(t)=te^-1

(b) y₂(t)= t-2

The table below shows a dataset representing the ages of employees working for three different districts. Assuming a minimum working age of 18 and a mandatory retirement age of 65:

1. What is the possible age range for each district?
2. What is the possible age range for all districts combined?
3. What is the computed age range for each district?
4. What is the computed age range for all districts combined?
5. What does the computed versus actual age range tell you about the dispersion computed in questions 1-4?
6. What is the standard deviation for age in each district?
7. What does the standard deviation tell you about the dispersion of age in each district?
8. What is the standard deviation of age in all districts combined?
9. What is the variance in age in each district?
10. What is the variance in age when all districts are combined?