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Q107. Our neighborhood association has a ten member board. Each year it plans to add four members. Write the difference equations that describe the size of the board (S) each year.

Q108. You are a small non-profit. Your sole funder says that each year it will double what you have as your balance at the end of the year. Each year you project spending 20,000 for programs. Ignore interest. Write difference equations describing your balance (B).

What special situations can you imagine we might get into? What, for example, happens if B0=$32,000? What happens if it is 50,000? 40,000? Q109. Each year the feral cat population grows by 3%. Let Cn be the number of cats n years from now. Assume there are presently 350. Write a difference equation that describes the cat population from year to year. Q110. Each year the feral cat population grows by 3%. Let Cn be the number of cats n years from now. Assume there are presently 350. Suppose that each year we catch and euthanize or place in homes 20 cats. Write the equations for this situation. Q111. Let's say we have a 2 year graduate program. The first year class is growing at a rapid rate 5% per year. Between the first and second years, 25% of the students change their minds or get jobs and leave the program. Among the second years, 10% leave before graduation. The program currently has 20 first year and 12 second year. Write difference equations to describe population in future years. Q114. A totalitarian country that prohibits migration has a birth rate b and a death rate d. How does the population change from one year to the next? We can interpret "birth rate" or "death rate" in two ways — as a rate, for example, deaths per 100 people — or as an absolute number, for example, 5 people per year. In general, we will mean the former. Let$P_{i}$be the population in year i. Write an expression for the population in year i+1. If the population in a given year is 100 and birth rate, b, is 5% and the death rate, d, is 6%, write out the population for the next 5 years. Q115. There are no births in a Shaker community, only R recruits per year. The death rate is d. What is the difference equation that describes this situation? Q116. The Eastville School Committee is agrees to an annual$200 per year salary increase for each Eastville teacher. Express this as a difference equation.

Q117. The Westtown School committee is more generous. It agrees to a 5.5% cost-of-living increase per year, plus a one time only $200 adjustment for past sins of omission. Express this as a difference equation. Q118. The Westtown School committee is more generous. It agrees to a 5.5% cost-of-living increase per year, plus a one time only$200 adjustment for past sins of omission. How much would it be worth to teachers if the one time adjustment were made before the first COLA rather than after.

Q119. Mills public policy program recruits R new students each fall. In the spring 0.0G (i.e., G%) students graduate. At the end of a typical year 0.0L (i.e., L%) of active students leave for personal or other reasons. Express the current student population, P, in terms of these figures.

Q120. My bathtub fills at 10 gallons per minute. It has a leak, though, whereby it loses 10% of it's volume per minute. It's a neat rectangular tub in which each 10 gallons is 2 inches of depth. How does it behave over time?

Q121. Consider a "leaky" reservoir. Current volume 1,200 million gallons. Inflow 200 million gallons per month. Consumption 150 million gallons per month. Leakage 5% of current volume per month.

1. Draw a stock and flow diagram of the situation.
2. Draw a causal loop diagram showing the relationship between reservoir volume and the net in/out flow.
3. Identify amount(s) and rate(s).
4. Write the difference equation in the form $P_{n+1} = a \times P_n + b$.
5. Calculate the expected equilibrium.
6. Set up Excel model.
7. Chart reservoir volume vs. time.

Q122. The most basic opportunity cost incurred when undertaking a project is the simple value of investing the money instead of spending it. A first step toward figuring out what that cost is is understanding compound interest. Show what happens to $1000 if the annual interest rate is 5%. Q125. Write an equation for the line passing through the points (3,2) and (0,5). Q136. Write out the difference equation that represents the following scenario and the first five terms of the corresponding sequence given the stated starting value. 1. Membership in a club goes up by 4 people each year. At year one it has 21 members. 2. A community's population increases by 4% each year. At year one it is 350. 3. A swimming pool, currently containing 100,000 gallons of water, is leaking at the rate of 2% per day but is being filled at the rate of 1,000 gallons per day. 4. A retirement account which stands at$120,000 earns 3% interest annually. The owner needs to withdraw \$1500 per month to pay for eldercare.

For each of these, graph Pn vs. time.

For each of these, graph Pn+1 vs. Pn

Q137. Derive the equation for the equilibrium value of a difference equation from a formula that shows how Pn+1 relates to Pn.

Q186. Our neighborhood Obama for America committee is an active one. It's so active that it wears people out. Over the course of the campaign it tends to recruit 4 new people every week but it also loses about 10% of its membership due to fatigue each week. The committee began in June with 6 members. Write the difference equations that describe the size of the committee (S) each week. What's the long term prognosis?

Q326. (A) Consider this plot of Pn+1 vs. Pn. Without worrying about what sort of system it might be, show that you understand how the chart works by describing the behavior of this system if it starts at time i at Pi=30. How about 70?

(B) If this is a model of attendance at, say, a protest rally with the axes representing percentage of the population, and Pn is how many showed up last week (a number everyone knows) and Pn+1 is how many that means we can expect this week (based on the distribution of individual thresholds - how many people need to be going for me to decide to go), how would you interpret the gaps A and B on the chart?

(C) Think about the "standing ovation model." What features does it add to the basic model described here.