Problem 1A Flexible Savings Account (FSA) plan allows you to put money into an account at the beginning of the calendar year that can be used for medical expenses. This amount is not subject to federal tax as you pay medical expenses during the year, you are reimbursed by the administrator of the FSA until the money is exhausted. From that point on, you must pay your medical expenses out of your own pocket. On the other hand, if you put more money into your FSA than the medical expenses you incur, this extra money is lost to you. The goal of this study is to identify the optimal amount of FSA contribution to maximize the expected salary after tax and medical expenses.Your annual salary is $70,000 and your federal income tax rate is 30%. Assume that you put $2,000 into your FSA at the beginning of the year and that medical expenses in a year are normally distributed with mean $2,000 and a standard deviation of $500.A static model is given in the IA5_Q1_FSA_template.xlsx. Note that the output is the amount of money left to you after i) putting money into the FSA, ii) paying tax, iii) paying out of pocket medical expenditure. Make sure you understand all the calculations in the model.Copy and paste your table. Based on the simulation results, answer the following questions.Problem 2The Greatyear Tire Company has developed a new tire and wants to jump start the marketing of this tire.Based on their testing of the product, it has mean lifetime of 36,500 miles and a standard deviation of 5,000 miles.They also determined that the lifetime follows a normal distribution with those parameters quite well.The marketing department decided on the following plan for promotion of the product:they are so sure that the product will wear well that they will promise to pay a rebate for any tire that needs replacement at a threshold.They are considering 5 thresholds:28,000, 29,000, 30,000, 31,000 and 32,000 miles, respectively.They will pay the purchaser back $1 for each full 100 miles the tire is below this threshold.(For example, if the threshold is 29,000, and the tire is replaced at 27,430 miles, the below threshold mileage is 1,570 miles (= 29,000-27,430).Then, the below threshold mileage is divided by 100 (that yields 15.7), so the number of FULL 100s below is 15.This person would be paid $15 for that tire.)The data files for both questions will be provided
Name: Section Number:
TA Name: Section Time:
Midterm Exam 2
May 18, 2005
Turn off and put away your cell phone.
You may use a calculator, but no other electronic devices are allowed on this exam.
You may use one page of notes, but no books or other assistance on this exam.
Read each question carefully, answer each question completely, and show all of your work.
Write your solutions clearly and legibly; no credit will be given for illegible solutions.
If any question is not clear, ask for clarification.
1. (4 points) For which value(s) of t are the vectors 3 − → i +2 → − j +t → − k and 2t − → i +4 → − j +t → − k
1 2 3 4 Σ
2. (4 points) Suppose that for some differentiable function g(x, y),
g(1, 2) = 4, gx(1, 2) = −3, gy(1, 2) = 5.
(a) Find the local linearization of g near x = 1 and y = 2.
(b) Approximate the value of g(1.02, 2.03).
3. (4 points) Find a vector that is perpendicular to the vectors 4 − → i + 3 → − j + − → k and
4 → − i + 6 → − j + → − k .
4. (6 points) Let f(x, y) = x2y5. At the point (−1, 2):
(a) Find a vector in the direction of maximum rate of change.
(b) Find a vector in the direction of minimum rate of change.
(c) Find a vector in a direction in which the rate of change is zero.
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