Calculation of half-life for alpha emission using time-independent
Schrodinger Equation using the following following information:
Radionuclide: 241-Am (Z=95); Ea = 5.49 MeV; Measured Half-Life~432y

Follow the steps involved and show your work for each subset question, not the
final answer:
(a) Evaluate the well radius [=separation distance (r) between the center of
the alpha particle as it abuts the recoil nucleus];
(b) Evaluate the coulomb barrier potential energy (U) for the well;
(c) Estimate the separation distance (r*) from the center of the potential well
where the coulomb potential equals the energy of the alpha particle;
(d) Assuming a square shaped single barrier of height “U”, evaluate the
tunneling probability by solving for the transmission coefficient and use of
the associated separation (=r*-r);
(e) Calculate the frequency with which the alpha particle strikes the well
boundary to try to get out of the well;
(f) Calculate the half-life and compare it with the known half-life for alpha
emission from 238U;
(g) Use the spatially-averaged effective approximation for the height of the
barrier by integrating the variation of “U” with distance from “r” to “r*” and
re-calculate the half-life and compare it with the known half-life for alpha
emission from 238U;
(h) Approximate the hyperbolic shaped barrier variation outside of the well
by breaking them up into 5 progressively reduced height square shaped
barriers, each with a width = (r*-r)/5 and calculate the tunneling
probabilities associated with each segment;
(i) Re-calculate for the half-life combining the probabilities from each of the 5
bins, and compare the value with the known half-life.

1. A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a par- ticular time, and let Y denote the number of hoses on the full service island in use that time. The joint pmf of X and Y appears in the accompanying tabulation.

a. WhatisP(X=1andY =1)?

b. ComputeP(X≤1andY ≤1).

c. Compute P(X ̸= 1 and Y ̸= 1).

d. Compute the marginal pmf of X and Y. Using pX(x), what is P(X ≤ 1)?

e. Are X and Y are independent? Explain.

2. Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable – X for the right tire and Y for the left tire, with joint pdf

?K(x2 +y2), 20≤x≤30, 20≤y≤30,

f(x,y) =

0, otherwise

a. What is the value of K?

b. What is the probability that both tires are underfilled?

c. What is the probability that the difference in air pressure between the two tires is at most 2 psi? (Hint: Draw the shaded region first and then find the boundary of the integration.)

d. Determine the marginal pdf of X and Y . e. Are X and Y independent? Explain.
f. Find E(X) and E(Y ).

Multiple Choice

1. Which of the following is not used to define a chamfer?

a. Angle and a distance

b. Distance and distance

c. Two angles and a distance

2. Which tool is used to draw a spring?

a. Coil

b. Loft

c. Sweep

3. The Edit Sketch tool can be applied to shapes created with which of the following tools?

a. Extrude

b. Rectangle

c. Revolve

d. Hole

4. The Edit Feature tool can be applied to shapes created with which of the following tools?

a. Circle

b. Line

c. Point, Center Point

d. Extrude

5. Which of the following parameters cannot be used to draw a work plane?

a. Angle to a plane

b. Point and a tangent

c. 3-Point

d. Tangent to face through

6. Which of the following is a material not listed under the Physical tab of the Properties dialog box?

a. Mild Steel

b. Aluminum Bronze

c. Glass

7. Sketched shapes can be projected between work planes using which tool?

a. Sweep

b. Boundary Patch

c. Move Face

d. Project Geometry

8. Which of the following will happen if a work plane is deleted?

a. The work plane will disappear from the screen and all entities will be deleted.

b. The work plane will disappear from the screen and all entities will remain in place.

9. Why are ribs used on molded parts?

a. To increase the part's flexibility

b. To balance the part

c. To increase the part's strength

10. The Face Draft tool is used to

a. Create airflow

b. Create an angled surface

c. Create a current behind a moving object

2. [1.1, 2.1, 2.2] In the recruitment process of a well-known Seattle hi-tech firm, résumés for a posted position are collected in an electronic drop box. An automatic sorting algorithm supervised by a junior recruiter whittles down the submitted résumés to an initial shortlist. All résumés in this shortlist go through the next three steps.

(1) Resource-type "junior HR person" checks the résumé for information about all non-technical aspects and ranks (fit ranking) all shortlisted candidates on a fit-to-firm scale. This takes about 8 minutes per résumé. (2) Resource-type "technical person" (usually with some help from the group that has posted the position) checks the résumé for evidence of technical skills and ranks the shortlisted candidates on a technical scale (technical ranking). This takes about 5 minutes per résumé. (3) There are two resources of resource-type "senior HR executive" who are charged with conducting preliminary phone conversation of about 24 minutes with each shortlisted candidate. That is, each shortlisted candidate get to have a conversation with any one of the two senior HR executives and the conversation takes 24 minutes.

To define the process boundary, we focus on three steps described above that are performed on all shortlisted candidates. Flow unit is shortlisted résumé.

(a) What is the capacity of each resource-type and capacity of the process in units per hour? Which resource-type is the bottleneck?

(b) What will be your suggestion (in words) to improve the capacity of the process? Going beyond your suggestion, if we can somehow manage to completely balance the process by redistributing work between different resource-types, what is the maximum possible capacity of the process?

1. Which of the following does not pose a risk to a person’s ability to service their required debt payments?
   1.      Variability in cash flows caused by changes in earnings
   2.     A decrease in asset values
   3.      Risks covered by insurance

               None of the above affect a person’s ability to service their required debt payments

A financial institution owns a portfolio of options dependent on the US dollar–sterling exchange rate. The delta of the portfolio with respect to percentage changes in the exchange rate is 6.1. If the daily volatility of the exchange rate is 0.5% and a linear model is assumed.

The estimated 10-day 99% VaR is $ .

Current models of climate change and cost-benefit analysis can rely heavily on the inclusion of extremely low probability but very high cost (catastrophic and irrevocable ecosystem changes, for example) events. Do inclusion of these events increase the validity of the analysis or do they bias the results?

Explain what might happen to accident rates if compulsory auto liability insurance laws were repealed and tort liability were restricted without mandating the purchase of PIP coverage. How could the government promote greater driving safety if these changes were made?

Write the balanced equation for the reaction between MnO4^- and C2)4^2-. Identify the oxidizing reagent and the reducing reagent. Identify the elements that undergo oxidation and reaction including their oxidation number changes. 2MnO4^- + 5 C2O4^2- + 16H^+ --> 2Mn^2+ + 10CO2 + 8H20