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
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?
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
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
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
= 1.22019004 − 1
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
|Due By (Pacific Time)||12/05/2012 09:00 am|
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