The refractive index of a transparent material can be determined by measuring the critical angle when the solid is in air. If θ_c= 41.5° what is the index of refraction of the material?

A light ray strikes this material (from air) at an angle of 37.1° with respect to the normal of the surface. Calculate the angle of the reflected ray (in degrees).

Calculate the angle of the refracted ray (in degrees).

Assume now that the light ray exits the material. It strikes the material-air boundary at an angle of 37.1° with respect to the normal. What is the angle of the refracted ray?

The fully-developed, steady,Poiseuilleflow of a liquidina horizontal, circular cylinderwith no swirl (i.e., Vθ= 0)under certain simplifying assumptions is given by:

Vz = (−∆???)4?(R2 –r2)Vr = 0Vθ= 0Here, Ris the inner radius of the pipe and r, θand zare the (cylindrical) coordinates of any point inside the pipe.Apply the continuity equation in cylindrical coordinates (Appendix B.8 of the text), given by()()()0zr1rrr1t=+++zrVVVto this flow (giving reasons why you drop each term) and obtain an ordinary differential equation for Vr. Integrate your solution and apply appropriate boundary conditions to obtain Vr.

Q 5.

Reagan has accumulated $3200 in savings and wishes to invest this money sensibly. The types of investments and their corresponding percentages, recommended by a financial advisor, are shown in the following pie chart. Find the amount of money Reagan should invest in bonds. Round your answer to the nearest whole dollar

Recommended Categories of Investment

Bonds 38.28%

Stocks 15.74%

Real Estate 10.22%

Mutual Funds 10.54%

Annuities 25.22%

Q 6.

Consider the following frequency table representing the distribution of cost of a paperback book (in dollars).

Step 1 of 2:

Determine the cumulative frequency for the second class.

Step 2 of 2:

Determine the cumulative frequency for the fourth class.

Q 7.

Consider the following frequency table representing the scores on a test.

Copy Data

Step 1 of 5:

Determine the lower class boundary for the fifth class.

Step 2 of 5:

Determine the upper class boundary for the second class.

Step 3 of 5:

Determine the class width of each class.

Step 4 of 5:

Choose the interval that contains the score, 57.9

Step 5 of 5:

Determine the number of scores between 39.5 and 55.5

2. The growth rate of a population of bacteria is directly proportional to the population p(t) (measured in millions) at time t (measured in hours).

(a) Model this situation using a differential equation.

(b) Find the general solution to the differential equation.

(c) If the number of bacteria in the culture grew from p(0) = 200 to p(24) = 800 in 24 hours, what was the population after the first 12 hours?

3. Find the particular solution y(x) to the following boundary-value problem, y ′′ + 9y = 0, with y(0) = −1 and y(π/6) = 1.

4. Consider the differential equation y ′ + 3t 2 y = e −t 3 with initial condition y(0) = 1. Find the particular solution of the differential equation.

6. Consider a rod: 0 < x < L with insulated sides. The temperature at the sides is fixed at 10 degrees C. At time t = t0, the rod is given an initial temperature distribution of f(x) degrees C, for 0 < x < L. Let u(x, t) denote the temperature at the point x on the rod at time t. The heat flow is modeled by the heat equation, ut − 2uxx = 0.

(a) Write the initial condition(s) of the problem in terms of u(x, t).

(b) Write the boundary condition(s) of the problem in terms of u(x, t).

(c) What does the solution of this problem represent for the rod?

(d) Suppose you are not interested in solving the problem but in finding the steady-state solution. What differential equation would you solve instead?

Given the data on scores of students final grade in statistics (in percent) determine the following statistics.

43 45 48 51 53 54 57 59 60 60 60 60 61 70 70 71 71 72 72 72 75 76 76 79 81 81 83 85 87 88 88 89 89 91 92 93 96 98 98 99 100 101 101

Assume students are only allowed to transfer the class if they receive a grade of 70 % or above. Use this fact to create a binomial distribution for students that are able to transfer and students not able to transfer the class. Do this by finding the proportion of students that receive a grade of 70 or above (this will be the value p and then q = 1 - p).

a. Determine the mean and standard deviation using the binomial distribution formulas.

b. Determine the range of usual value by finding the values that are significantly low and significantly high.

c. Use a normal continuous distribution to APPROMATE the binomial discrete probability distribution to determine the probability that at least 30 students score at a 70 or more. (Be sure to use the boundary to get the more accurate/correct answer.) Show an approximation box to verify your boundaries.

d. Use a normal continuous distribution to APPROMATE the binomial discrete probability distribution to determine the probability that exactly 30 students score at a 70 or more. (Be sure to use the boundary to get the more accurate/correct answer. Show an approximation box to verify your boundaries.

4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula xn+1 , yn+1 = xn, yn − ε ∇f(xn, yn). In general, ε does not have to be a constant, but in this question, for demonstrative purposes, we set ε = 0.1. Let f(x, y) = 3.5x 2 − 4xy + 6.5y 2 and x0 and y0 be any real numbers. (a) For all x, y ∈ R compute ∇f(x, y) and find a matrix A such that [3] A x y = x y − ε ∇f(x, y). Write an expression for xn yn in terms of x0 and y0 and powers of A. (b) Find the eigenvalues of A. [1] (c) Find one eigenvector corresponding to each eigenvalue. [2] (d) Find matrices P and D such that D is diagonal and A = P DP −1 . [1] (e) Find matrices Dn , P −1 and An . Find formulas for xn and yn. [4] (f) Suppose x0 = y0 = 1. Find the smallest N ∈ N such that xN yN ≤ 0.05. [3] (g) Sketch the region R consisting of those (x0, y0) such that xN ≥ 0, yN ≥ 0 and [4] xN yN ≤ 0.05, xN−1 yN−1 > 0.05, where N is the number found in part (f). Write an equation for the boundary of R. Which points of the boundary belongs to R and which do not?

A 2012 New York Times article noted the growing national trend of gated communities: Across the United States, more than 10 million housing units are in gated communities, where access is “secured with walls or fences,” according to 2009 Census Bureau data. Roughly 10 percent of the occupied homes in this country are in gated communities... Between 2001 and 2009, the United States saw a 53 percent growth in occupied housing units nestled in gated communities. Over the past thirty years, residential income segregation has also increased, with more upper-income households located in majority upper-income neighborhoods. Drawing on Kendall’s (2006) study about elite practices of boundary maintenance, explain how residential segregation contributes to class-based inequality.A 2012 New York Times article noted the growing national trend of gated communities: Across the United States, more than 10 million housing units are in gated communities, where access is “secured with walls or fences,” according to 2009 Census Bureau data. Roughly 10 percent of the occupied homes in this country are in gated communities... Between 2001 and 2009, the United States saw a 53 percent growth in occupied housing units nestled in gated communities. Over the past thirty years, residential income segregation has also increased, with more upper-income households located in majority upper-income neighborhoods. Drawing on Kendall’s (2006) study about elite practices of boundary maintenance, explain how residential segregation contributes to class-based inequality.

You opened an account and deposited X Dollars on January 1, 2002 in National City Bank. Any balance in the account will earn 5% per year. You withdrew $500 on January 1, 2006 and $500 on January 1, 2008. You closed out this account on January 1, 2011 and received $700. How much did you initially deposit (X) in National City at the time you opened the account?

The U.S. federal ban on assault weapons expired in September 2004, which meant that after 10 years (since the ban was instituted in 1994) there were certain types of guns that could be manufactured legally again. A poll asked a random sample of 1,200 eligible voters (among other questions) whether they were satisfied with the fact that the law had expired. The datafile linked below contains the results of this poll (Data were generated based on a poll conducted by NBC News/Wall Street Journal Poll). We would like to estimate p, the proportion of U.S. eligible voters who were satisfied with the expiration of the law, with a 95% confidence interval.

M.E.= 2.9%

(a) How many of the 1,200 sampled voters were satisfied?

(b) What is the sample proportion (p-hat) of those who were satisfied?

(c) What is the 95% confidence interval for p? Interpret this interval.

In question 1 you found that the margin of error of this poll was about 2.9%. What is the margin of error of the confidence interval you found in question 2?

In 2003/2004, three administrations ago, Vision 20-20 – a strategy for development and
progress, aimed at bringing Nigeria among the top 20 industrialized nations in the world
by 2020 A.D. was drafted. A positive challenge to entrepreneurial and functional education
in Nigeria. The vision sets out a 7-point Agenda that will drive the process of achieving
national objectives. It consists of Goals, Macroeconomic framework, Financial and plan
implementation strategies, to fast track the policies. With the aid of a well-labeled chart,
show the “NEEDS” at a glance.