3) Now you will plan for your retirement. To do this we need to first determine a couple of values.

a. How much will you invest each year? Even $25 a month is a start ($300 a year), you’ll be surprised at how much it will earn. You can choose a number you think you can afford on your life circumstances or you can dream big. State what you will use for P, r, and n to earn credit. (3 points)

The typical example of a retirement investment is an I.R.A., an Individual Retirement Account, although other options are available. However, for this example, we will assume that you are investing in an I.R.A. (for more information see: http://en.wikipedia.org/wiki/Individual_Retirement_Account ) earning 8% interest compounded annually. (This is a good estimate, basically, hope for 10%, but expect 8%. But again this is just one example; I would see a financial advisor before investing, as there is some risk involved, which explains the higher interest rates.) List your P, r, and n to earn points for this question.

b. Determine the formula for the accumulated amount that you will have saved for retirement as a function of time and be sure to simplify it as much as possible. You need to be able to show me what you used for r, n, and P so that I can calculate your answers. Plug in those values into the formula and simplify the equation. (5 points)

c. Graph this function from t = 0 to t = 50. (6 points) Ways to show graphs: ? Excel ? Hand draw, take a pic with phone and import it into your document as a picture. ? Online graphing calculator program (try googling free graphing calculators or use one listed in the Tech websites from Module 1)

d. When do you want to retire? Use this to determine how many years you will be investing. (65 years old is a good retirement-age estimate). You need to say how old you are (Age 29) if you are retiring when you are 65 or tell me how long until you retire. State what you will use for t. (2 points)

e. Determine how much you will have at retirement using the values you decided upon above. (5 points)

f. How much of that is interest? (4 points)

g. Now let’s say you wait just 5 years before you start saving for retirement, how much will that cost you in interest? How about 10 years? How about just 1 year? (10 points) Now you need to consider if that is enough. If you live to be 90 years old, well above average, then from the time you retire, to the time you are 90, you will have to live on what you have in retirement (not including social security). So if you retired at 65, you will have another 25 years where your retirement funds have to last.

h. Determine how much you will have to live on each year. Note, we are neither taking into account taxes nor inflation (which is about 2% a year). (5 points) Let’s look at this from the other direction then, supposing that you wanted to have $50,000 a year after retirement.

i. How much would you need to have accumulated before retirement? (5 points)

j. How much would you need to start investing each year, beginning right now, to accumulate this amount? A “short-cut” to doing this is to first compute the effective yield at your retirement age, then divide this amount into Part (i). This is the amount you well need to invest each year. (5 points)

k. That was just using $50,000, how much would you want to have each year to live on? Dream big or reasonable depending on your occupation! Now using that value, repeat parts (i) and (j) again. You need to state what you would want to live on and it needs to be something besides $50,000. (10 points)

Your answer to (k) would work, if you withdrew all of your retirement funds at once and divided it up. However, if you left the money in the account and let it draw interest, it is possible that the interest itself would be enough to live on, or at the very least if you had to withdraw some of the principle, the remaining portion would still continue to earn interest. Essentially, what you have found is the upper bound for the amount of money that you will need to invest each year to attain your financial goals.

l. Finish by summarizing what you have learned in the entire project and consider setting a goal towards saving for retirement. (Your answer should be in complete sentences free of grammar, spelling, and punctuation mistakes.) This should be a paragraph not one sentence. (10 points)

Just because two or more values are different does not mean that they are different in a statistically significant manner. Researchers rely on the p values that are generated for each of their statistical tests to determine significance. If the p value is larger than the alpha, then they are not different in a statistically significant manner, and therefore the values are not considered different. In this journal activity, consider these concepts in terms of the differences between null and alternative hypotheses, and discuss this question: What is the difference between failing to reject the null hypothesis and having evidence to support the alternative hypothesis?

Complete the table to determine the balance A for P = $1200 dollars invested at rate r = 6% for t = 10 years and compounded n times per year. (Round your answers to two decimal places.)

Argument A

Since all triangles have three sides, and an isosceles triangle is a triangle, it follows that an isosceles triangle has three sides.

Argument a is:

inductive/deductive;

valid/invalid/strong/weak;

sound/unsound/cogent/uncogent?

Argument B

There is a large crack in Philadelphia's Liberty Bell. From this, we can conclude that there was a defect in the bell's craftsmanship.

Argument B is:

inductive/deductive;

valid/invalid/strong/weak;

sound/unsound/cogent/uncogent?

Argument C

Paleontologists now agree that the Tyrannosaurus Rex was actually a peaceful, plant-eating dinosaur. Therefore, it is probably true that the Tyrannosaurus Rex was not a meat-eater.

Argument C is:

inductive/deductive;

valid/invalid/strong/weak;

sound/unsound/cogent/uncogent?

What does a sampling distribution have to do with a confidence interval?

Maximization applications

A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and $20 of capital, and each acre of corn requires 16 hours of labor and $40 of capital. The farmer has at most 800 hours of labor and $2400 of capital available. If the profit from an acre of wheat is $80 and from an acre of corn is $100, how many acres of each crop should she plant to maximize her profit?

Consider the following scenario:

A drum used for percussion has sides modeled by a hyperbola. Suppose the total height of the drum is known. The diameter at the top of the drum is 16 inches, and the minimum diameter is 7 inches, occurring at a height of 9 inches. Assume that the center of the hyperbola occurs at the height where the diameter is the least.

Choose your own value for the height of the drum based on the scenario above and address the following:

1. Draw a graph or figure to represent this situation.
2. Describe how the concepts from this module can be applied in this case.
3. Determine the equation of a hyperbola that models the sides of the drum.
4. Provide another example of a scenario that involves the same concept.

DISCUSS UNDERSTANDING HOW TO SOLVE AN EQUATION ALGEBRAICALLY AND GRAPHICALLY.

A farmer raises only sheep and geese. She wants to raise at least 16 animals, including 7 sheep. She spends $34 to raise a sheep and $15 to raise a goose, and she has $1020 availble for this project. Write a system of inequalities that describe all the given conditions. Graph the feasible region.

Compound Interest In Exercises 17 and 18, determine the time necessary for $1000 to double if it is invested at interest rate compounded (a) annually, (b) monthly, (c) daily, and (d) continuously

17. r = 11%

18. r = 10 1/2%

Translate the phrases to a system of linear equations: y= a1*x+b1 and y=a2*x+b2 with a variables, e.g. x and y. What do the y-intercepts represent in your example? What does the solution (or intersection) represent in your example? Solve the system of equation for x and y.

b. Find three rational numbers between 1/9
and 0. 12 overbar

Let A be some m*n matrix. Consider the set S = {z : Az = 0}. First show that this is a vector space. Now show that n = p+q where p = rank(A) and q = dim(S). Here is how to do it. Let the vectors x₁, . . . , x_p be such that Ax₁, . . . ,Ax_p form a basis of the column space of A (thus each x can be chosen to be some unit vector with a 1 corresponding to the position of a column vector that is part of a (maximally) linearly independent set) and let the vectors z1, . . . , zq be a basis for S. Then show that the two sets together i.e. the set {x1, . . . , xp, z1, . . . , zq} form a basis of the n-dimensional Euclidean space.

Using the result above, offer a direct proof of the result r(X′X) = r(X) without appealing to the product rank theorem

axis symmetry, graph max or min value, domain, range and all intervals where function increases or decreases. f(x)=5(x-3)^2+6 f(x)=-7(x-8)^2+1 f(x)=-6(x+4)^-8 f(x)=2(x+2)^2+9

A rectangular piece of metal is 30in longer than it is wide. Squares with sides 6in long are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 2706incubed3​, what were the original dimensions of the piece of​ metal?

what is the original width?

what is the original length?