You may need me to attach the original document to see the graphs, however I think the problems can be solved without them.

NOTE** The homework helper HAS to hand write out how you got the answer and scan those files back to me** Please let me know in your response that you understand that. Also let me know your price!

Suppose that there is an energy source, like coal, for which we have some finite

reserve, say 2500 million tons. Further assume that at the current yearly rate of

consumption, the reserve will be completely exhausted at the end of some specific

time interval, say 500 years. A graph indicating the amount in reserve would look

like this.

1. In this example, what is the indicated consumption rate per year?

2. Create a linear function showing the amount of coal in reserve R (in millions

of tons) at time t (years from now). (So, R is a function of t .) Verify that

your function is consistent with the data that at time t = 0 , R = 2500 and at

time t = 500 , R = 0 .

The model that you just created assumes that there is no change in the amount of

coal consumed each year. How would this model change if the amount of coal

consumed each year actually grows? How long will the reserves last in such a case?

The rest of the project will provide a guide to creating such a model.

According to the presentation, with a growth in the consumption of coal as small as

1% per year, the reserve will be exhausted in approximately 180 years. With a

growth in consumption of 3% per year, it will be exhausted in less than 100 years.

See the graph here.

3. In item 1 above, you determined the current yearly consumption rate. If this

amount is growing 1% each year, create an exponential model showing the

amount of coal consumed C in time t years from now. According to your

model, what will the yearly consumption of coal be in 5 years? 10 years?

100 years?

4. To determine at what point the reserves will be exhausted, it is not enough

just to know how much coal is consumed in a given year, but how the usage

from year to year is adding up. If we use 10 million tons one year and 11

million tons the next, then we have exhausted 21 million tons from the

reserve.

a. By adding together the amount of coal consumed in each of the first

five years (so C(0) + C(1) + C(2) + C(3) + C(4) ), determine the

amount of coal left in the reserves at the end of 5 years at this 1%

growth rate. How does this compare to the amount left in the reserves

at the end of 5 years when there is no growth in use?

b. Now determine the amount of coal left in the reserves at the end of 10

years at this 1% growth rate. How does this compare to the amount

left in the reserves at the end of 10 years when there is no growth in

use?

5. Although not impossible to do, determining the usage in each individual year

and adding all those numbers together is a tedious process. Fortunately,

there is a formula in mathematics that can help us simplify the calculation.

1+ a + a2 + a3 + a4 + ...+ an = an+1 − 1

a − 1

So, for example,

1+ 1.01+ 1.012 + 1.013 + 1.014 + ...+ 1.0119 = 1.0120 − 1

1.01− 1

= 1.22019004 − 1

0.01

= 22.019004

a. Use this formula to find the total amount of coal consumed over the

first 5 years at this 1% growth rate. Is this answer consistent with

what you got in 4a above? Do the same for the first 10 years and

verify that it is consistent with figures in 4b above.

b. Find a general formula for the total amount of coal consumed over the

n years at this 1% growth rate. This use the formula to find a function

that shows the amount of coal remaining in the reserves A at the end

of n years.

6. Now that you have a function, find the value of n at which point the coal

reserves will be completely exhausted. (So, A(n) = 0 .) Hint: You should use

the tool of logarithms to answer this question.

7. How will the function change if coal consumption grows at 3% per year? Use

this modified function to determine when the coal reserves will be exhausted

in the 3% growth per year case.

8. Briefly discuss the relevance that comparing such models has to the

discussion of sustainability. Should we expect the yearly demand on

resources like coal to stay constant? What additional challenges are there on

the quest for a sustainable world when the notion of exponential growth is

included?

Subject | Mathematics |

Due By (Pacific Time) | 12/05/2012 09:00 am |

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