Please show all work. In the coming year a country plans to consume 8000 units o

Question

Please show all work.

In the coming year a country plans to consume 8000 units of a resource, and in each year thereafter consume 2% less than in the previous year. (a) Write the first four terms of a series that gives the country's total future consumption of the resource. __________ +_________ +__________ +_________ + middot middot middot (b) Find the sum of the series you wrote down in part (a). (c) Suppose the country finds out that it must limit its future consumption to 320,000 units. If the country is committed to using 8000 units in the coming year, and will reduce its yearly consumption by a fixed percentage each year thereafter, what must this percentage be? (d) Now suppose the country must limit its future usage to 320,000 units and will reduce its consumption by 2% yearly. How many units can the country consume in the coming year? (e) Assuming that the country uses 8000 units in the coming year, create a table that shows the total future consumption T as a function of the annual reduction in usage d (note that the first entry is simply your answer to part (b)). d T 2% 3% 4% 5% 6% (f) Determine the correct responses: T is an increasing/decreasing function of d, and the graph of T as a function of d is concave up/concave down.

Explanation / Answer

a) Each year 2% less than previous year

Hence I year = 8000

II year =8000(98%) = 7840

III year = 8000(0.98^2) = 7683.20

III year = 8000(0.98^3)=7529.54

Iv year = 8000(0.98^4)=7378.95

...

b) Sum of the series = 8000(1+0.98+0.98^2+...)

= 8000 (1/(1-0.98)) = 8000 /0.02

= 400000

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c) Here S = sum of infinite terms = 320000

i.e. 8000(1/1-r) = 320000

1/1-r =40

40-40r =1

40r = 39

r = 0.975

i.e. reducing at the rate of 0.025 or 2.5%

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