In a slowly cooled hypereutectoid iron-carbon steel,
the pearlite colonies are normally separated from each
other by a more or less continuous boundary layer of
cementite. Explain how this microstructure develops. Use
simple sketches to illustrate your answer

Solve the following boundary value problem by Laplace Transform. (If you solve with another
method you will NOT get credit. There is nothing wrong about the conditions.)
d2y
dt2 + y = cos(2t); y0(0) = 0; y0(

2
) = ?1:

Imagine an octagon(8 vertices) with all its diagonals. Is it possible to draw this without lifting your pen (and no backtracking). If we attach a triangle for each boundary edges of the Octagon, can you draw this without lifting your pen? Explain

Let ?be the solid region bounded by the surfaces ?=10−?2−?2and ?=1. Let ?be the boundary of ?. If ?⃗=<?,?,?>, compute the total flux over ?in two ways:(a)Directly as a surface integral.Include a sketch of ?in your answer (b)As a triple integral using the Divergence Theorem

Find the solution to the boundary value problem:

?^2?/??2−6??/??+5?=0,   ?(0)=10,?(1)=6

Please show me all your work, I want to know where I went wrong. Thank you!

Find the eigenfunctions for the following boundary value problem.

x²y?? ? 15xy? + (64 + ?)?y ?=? 0,    y(e^?1) ?=? 0, ?y(1) ?=? 0.

In the eigenfunction take the arbitrary constant (either c₁ or c₂) from the general solution to be 1

Compare the maximum thickness of the boundary layer that develops due to uniform flow past a flat plate when the fluid is

(a) air, (b) water, (c) glycerol.

The plate length is 3 m and the approaching fluid velocity is 4 m/s.

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε).

(a) Show that every element of N∗(x;ε) is an interior point.
(b) Determine the boundary of N∗(x;ε) and prove your answer is correct.

1.   A researcher conducted a survey on a university campus for a sample of 64 seniors. The research reported that seniors read an average of 3.12 books in the prior academic semester, with a standard deviation of 2.15 books. Determine the probability that the sample mean is:
   a) above 3.45
   b) between 3.38 and 3.58
   c) below 2.94
2.   Al Gore, the Democratic candidate for President of the USA in the 2000 election, believes that the proportion of voters who will vote for a Democrat candidate in the year 2004 presidential elections is 0.65. A sample of 500 voters is selected at random.
a.   Assume that Gore is correct and p = 0.65. What is the sampling distribution of the sample proportion ? Explain.
b.   Find the expected value and the standard deviation of the sample proportion .
c.   What is the probability that the number of voters in the sample who will vote for a Democrat presidential candidate in 2004 will be between 340 and 350?
3.   A random sample of 10 waitresses in Iowa City, Iowa revealed the following hourly earnings (including tips):

$19   18   15   16   18   17   16   18   20   14

If the hourly earnings are normally distributed with a standard deviation of $4.5, estimate with 95% confidence the mean hourly earnings for all waitresses in Iowa City.
4.   The number of cars sold annually by used car salespeople is normally distributed with a standard deviation of 15. A random sample of 400 salespersons was taken and the mean number of cars sold annually was found to be 75. Find the 95% confidence interval estimate of the population mean. Interpret the interval estimate.
5.   Suppose that in a large city, the annual income of real estate agents is normally distributed with a standard deviation of $7,500. A random sample of 16 real estate agents reveals that the mean annual income is $52,000. Determine the 99% confidence interval estimate of mean annual income of all real estate agents in the city.
6.   A medical statistician wants to estimate the average weight loss of people who are on a new diet plan. In a preliminary study, he guesses that the standard deviation of the population of weight losses is about 10 pounds. How large a sample should he take to estimate the mean weight loss to within 2 pounds, with 90% confidence?

Explain what it means to be ethical as it relates to personal, academic, and professional growth.