A hydrogen-filled balloon is to expand to a sphere 20 m diameter at a height of 30 km where the absolute pressure is 1100 Pa and the temperature is – 40 degree C. If there is to be no stress in the fabric of the balloon, what volume of hydrogen must be added at ground level where the absolute pressure is 101.3 kPa and the temperature is 15 degree C?

Decidability (6 + 6 + 6 + 6 credits) For each of the following problems, state whether it is decidable. If it is decidable, outline a decision procedure for it. If not, prove that it is undecidable. In each case, the input is the code (M) of a Turing machine M = (0,2,1, 8,90, B, F).

(a) Is there an input x € * of length < 10 such that M halts in < 100 steps when run on x?

(b) Is there an input 2 € * of length > 10 such that M halts in <100 steps when run on x?

(c) Is there an input x € * of length < 10 such that M runs for > 100 on input x?

(d) Is there an input x * of length > 10 such that M halts in > 100 steps when run on z?

Innis consulting group Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs and provides an annual rate of return of $5 per dollar invested ; each unit of the money market fund costs $100 and provides an annual rate of return of $4 for each dollar invested. The client wants to minimize risk subject to the requirement that the annual income from the investment is at least $60,000. According to Innis’s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis’s client also specified that at least $300,000 be invested in the money market fund.
a. Determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio.
b. How much annual income will this investment strategy generate?
c. Suppose the client desires to maximize the annual return. How should the funds be invested?

SCENARIO
It’s been eighteen months since he took over the Des Moines office, and Doug Bloom is finally feeling like he has the job under control. His office has exceeded their sales quotas for each of the last 5 quarters and profits are improving. He also felt like he understood himself better and that he had grown as an individual. At dinner the other night, he mentioned to his wife, “You know, as you get involved in this job, your personality changes. Some managers become very people-directed and others are more sales- and customer-directed. I doubt that you find many people who are perfectly balanced on these dimensions. I think I’m leaning toward the people side now, which surprises me. I always loved the customer contact so much when I was selling.”
The next day at work he received a call from Vinny Raccioppi, Vice President of Marketing and Sales for D & M, informing him that as part of his management development he was given a special assignment. Vinny would like to meet with him in New York the following week to discuss his assignment.
In New York, Doug is informed that his special assignment is to plan for the phasing-out of independent insurance brokers in the mountain states of Montana, Idaho, Wyoming, Utah, and Colorado. D & M intended to hire full-time salespeople to cover these geographies. Among other reasons for the change, the population in these areas is growing and the company feels that it can get better penetration of the market with its own dedicated sales force. Vinny also mentions that he believes the sales agents have grown “older and wealthier” and are losing the “energy and drive” of earlier years.
D & M currently has two regional sales managers responsible for managing the brokers in these states. The regional managers are responsible for recruiting high-quality brokers, working with them to ensure that understand and know how to sell D & M’s products, and producing the sales volumes quotas in these areas. The sales manager will need their help to put together an in-house sales force and to undertake damage control when the sales agents are informed of the new organization.
Doug’s reaction is one of surprise, followed by a realization of the magnitude of the undertaking. One of Doug’s immediate concerns was in controlling the damage to customer relationships that may arise when the brokers are informed of the new sales organization. Vinny informs Doug that he has two months to complete a plan for starting the new sales force. Doug says that he had better get started right away and suggests that he meet with the two regional sales managers this week.
The following week, Doug arranges to meet with the two regional sales managers. Doug informs the managers of the impending change in organization and of the necessity to develop a through and thoughtful plan of action. After expressing their surprise, the regional sales managers mention their concern with agents’ reactions. They both indicated that there is no way this plan can be kept a secret until it is implemented and that the brokers are likely to be angry about losing the 20 percent commissions they have become used to receiving. During this discussion, possible retaliatory actions by the brokers are discussed. At the conclusion of the meeting, the vice president suggests that they all think about how to “ease out” the 34 agents without suffering undue economic recriminations. They agree to get back together for a meeting the next week to discuss alternative plans.
KEY DISCUSSION QUESTION
1) How would you advise Doug to conduct the meeting with the two regional sales managers?

A candidate for mayor in a small town has allocated$40,000 for last-minute advertising in the days precedingthe election. Two types of ads will be used:radio and television. Each radio ad costs $200 andreaches an estimated 3,000 people. Each televisionad costs 0 and reaches an estimated 7,000 people.In planning the advertising campaign, the campaignmanager would like to reach as many peopleas possible, but she has stipulated that at least 10 adsof each type must be used. Also, the number of radioads must be at least as great as the number of televisionads. How many ads of each type should beused? How many people will this reach?