megannicole51
  • megannicole51
Throughout much of the 20th century, the yearly consumption of electricity in the US increased exponentially at a continuous rate of 7% per year. Assume this trend continues and that the electricity energy consumed in 1900 was 1.4 million megawatt-hours. (a) Write an expression for yearly electricity consumption as a function of time, t, in years since 1900. (b) Find the average yearly electrical consumption throughout the 20th century. (c) During what year was electrical consumption closest to the average for the century? (d) Without doing the calculator for part (c), how could you have
Mathematics
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katieb
  • katieb
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megannicole51
  • megannicole51
predicted which half of the century the answer would be?
anonymous
  • anonymous
of something increases exponentially. you are going to be working with something^(t) where t is your time in years. 7% increase means that one year later it is 107% of the last year. Try and find an exponent involving t that will increase your base by 7% when t is 1 (i.e 1 year later)
megannicole51
  • megannicole51
im sorry i dont really understand can you elaborate

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anonymous
  • anonymous
as Taplin44 said, you need an equation:\ y(t) = Ae^(bt) where y(0) = A (energy at 1900 which is given)
anonymous
  • anonymous
as for b, you want this factor e^b (t=1) to be 1.07 so that after 1 year you will have a grow of 107% 1.07A
anonymous
  • anonymous
so now you can find b : e^b = 1.07
anonymous
  • anonymous
sorry for my English.
megannicole51
  • megannicole51
do u mean the exponent b or the second question?
anonymous
  • anonymous
Yes, he did.
anonymous
  • anonymous
yes i did lol
anonymous
  • anonymous
Coolsector is so cool!
megannicole51
  • megannicole51
so then how would i do (c)?
anonymous
  • anonymous
you found the average ?
anonymous
  • anonymous
Yeah, did you find the average?
anonymous
  • anonymous
all you have to do is average = Ae^(bt) and now find t
anonymous
  • anonymous
You can do this! I believe in you1
megannicole51
  • megannicole51
how do i bring down the exponent? its been a long night of calc and my brain is fried
anonymous
  • anonymous
you mean how you find b using e^b = 1.07 ?
megannicole51
  • megannicole51
yes
anonymous
  • anonymous
take ln from both sides
anonymous
  • anonymous
Yes, Coolsector is correct, he is never wrong!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
megannicole51
  • megannicole51
asdfasdf54514....get a life.
megannicole51
  • megannicole51
b=ln1.07
anonymous
  • anonymous
correct
anonymous
  • anonymous
LOL... see, he doesn't lie!!!!!!
megannicole51
  • megannicole51
so thats the average?
anonymous
  • anonymous
Block me please. :) Actually never mind, I will leave.How does one untag himself?
anonymous
  • anonymous
no no
anonymous
  • anonymous
now we are only working on finding the expression for (a)!
anonymous
  • anonymous
im not talking about (b). my bad for calling the parameter b
anonymous
  • anonymous
but it is only part of the expression
terenzreignz
  • terenzreignz
@Coolsector a question, do we really have to go through the part where we have \[\Large Ae^{bt}\] with the constant e instead of just using \[\Large A(1.07)^t\]?
anonymous
  • anonymous
no
megannicole51
  • megannicole51
im kinda confused
terenzreignz
  • terenzreignz
Sorry @megannicole51 I was only asking Coolsector... don't worry, you guys seem to be on the right track :D
megannicole51
  • megannicole51
oh okay lol then keep going I'm super tired and this is my last problem! :)
anonymous
  • anonymous
so ill try again. we are now looking for the expression they ask in (a): we want exponential expression of the form: y(t) = Ae^(Bt) y(t) will tell us the energy consumption at time t in years after 1900 so that y(0) = A will be the energy consumption at 1900!
anonymous
  • anonymous
so can you tell me what A is ?
megannicole51
  • megannicole51
Y(0)?
anonymous
  • anonymous
y(0) = the energy consumption at the year 1900
megannicole51
  • megannicole51
im not sure what A is then
anonymous
  • anonymous
y(0) = A = the energy consumption at the year 1900 it is given in the question
megannicole51
  • megannicole51
so i was right?
anonymous
  • anonymous
where ?
megannicole51
  • megannicole51
u asked what A was
anonymous
  • anonymous
so what is it ?
megannicole51
  • megannicole51
idk.
anonymous
  • anonymous
again, the expression we are working on y(t) = Ae^(bt) y - the energy consumption since 1900 (when t=0 it tells us the energy consumption at 1900 when t =1 it tells us the energy consumption at 1901 etc)
anonymous
  • anonymous
so y(0) = A and it is given in the question : 1.4 million megawatt-hours
anonymous
  • anonymous
so now we have: y(t) = 1.4e^(bt) and it satisfies our first condition : y(0) = 1.4
anonymous
  • anonymous
(y in units of million megawatt-hours)
anonymous
  • anonymous
understand so far ?
kropot72
  • kropot72
The equation for (a) is: \[Yearly\ consumption=1.4e ^{0.07t}\] (b) \[Average\ thru \ 20th\ century=\frac{\int\limits_{t=0}^{t=100}1.4e ^{0.07t}.dt}{100}\]
megannicole51
  • megannicole51
okay go on
anonymous
  • anonymous
well i dont understand what is the point in giving the answers but whatever
kropot72
  • kropot72
Notice that using the equation for (a) the growth after 1 year is 7.251% not 7%, the reason being that the growth is stated to be exponential and continuous. The formula is derived using calculus. Can you evaluate the definite integral in the equation for (b)?
kropot72
  • kropot72
@Coolsector You posted "..........to be 1.07 so that after 1 year you will have a grow of 107%". Note that after 1 year the consumption has grown to 107.251% of the initial consumption. I intervened to try to clarify matters.
anonymous
  • anonymous
if we could keep all the numbers of b from e^b = 1.07 we would get a growth of 7%
anonymous
  • anonymous
the values of y for t = 1,2,3,4.. would be accurate if we could.
kropot72
  • kropot72
@Coolsector You are confusing the "continuous growth rate of 7% per year" with the annual growth and expecting the consumption after 1 year to be 107% of the initial consumption. However this expectation is not correct. You will find a good explanation of exponential growth here. http://math.ucsd.edu/~wgarner/math4c/textbook/chapter4/expgrowthdecay.htm
anonymous
  • anonymous
i understand what you are trying to say. but still every year the consumption grows at 7% for t not integer we will get different numbers but when t is integer we will get every time growth of 7% since it is like "the end of the year" and i think there is nothing to do with that 107.251% that you talked about. this is because of a different reason - the reason that you rounded up b to 0.07.
anonymous
  • anonymous
anyway this is not really matters here.
kropot72
  • kropot72
You are confusing 'Discrete exponential growth' and 'Continuous exponential growth'. The question deals with continuous exponential growth. The difference certainly does matter here. The derivation of the formula for continuous exponential growth is given here. http://mathworld.wolfram.com/ExponentialGrowth.html
anonymous
  • anonymous
im not confusing anything. the way i was solving it is correct (this is what you did in fact)
anonymous
  • anonymous
the expression we got is continuous but it still give the right values for every year
kropot72
  • kropot72
What expression did you get?
anonymous
  • anonymous
i wrote up there "you need an equation:\ y(t) = Ae^(bt) where y(0) = A (energy at 1900 which is given) as for b, you want this factor e^b (t=1) to be 1.07 so that after 1 year you will have a grow of 107% 1.07A so now you can find b : e^b = 1.07"
anonymous
  • anonymous
continuous expression can be still correct for specific discrete values. such as here.
kropot72
  • kropot72
But your value for b is incorrect. The growth rate is 0.07 per year, so b must equal 0.07. You have contrived a value for b to make the consumption after 1 year be 107% of the initial consumption. That is incorrect and you are not following the theory in the link that I posted.
anonymous
  • anonymous
my value for b is 0.067658648.. which gives e^b = 1.07 which is the exact value
anonymous
  • anonymous
because you rounded b into 0.07 you got a higher precent
anonymous
  • anonymous
there is nothing we can do about it - we have to round. but this is the reason.
kropot72
  • kropot72
No, I did not round the value. I used the decimal equivalent of 7% as given in the question.
anonymous
  • anonymous
ok
kropot72
  • kropot72
As explained in the link, continuous compounding as used in finance is basically the same. In the case of annual compounding at an annual interest rate of 7% a $100 initial investment is worth $107 after the first year. However with continuous compounding at an annual interest rate of 7%, the amount after 1 year is \[A=100e ^{0.07}=$107.251\]
anonymous
  • anonymous
well i still dont understand why you think determining 0.07 is better than finding this 0.067.. if i look at the examples at the link you gave they do what i did
anonymous
  • anonymous
but really i dont care for it. rounding this number would give me 0.07 so i would go with it anyway
anonymous
  • anonymous
anyway. look at higher numbers like if the growth was 60% what would you do then ? say that the factor is 0.6?! this is completely wrong
anonymous
  • anonymous
for 60% growth the factor should be about 0.47
anonymous
  • anonymous
please answer for my last two comments as well.
kropot72
  • kropot72
Your reasoning is incorrect. You need to understand the ordinary differential equation that leads to the formula for exponential growth. You are taking a figure for growth and trying to fit it to discrete values of time (years). Taking 60% growth in consumption we get: \[1.4\times1.6=1.4e ^{0.07t}\] Solving for t gives t = 6.714 years Note that fractions of a year are valid for continuous exponential growth.
anonymous
  • anonymous
no i said instead of 7% growth 60% growth in one year. find the factor b then.
anonymous
  • anonymous
according to you it should be 0.6 which cant be right
kropot72
  • kropot72
The growth factor remains constant when applying the equation for (a) in the question: \[Yearly\ consumption=1.4e ^{0.07t}\]
kropot72
  • kropot72
You posted "according to you it should be 0.6" I made no such statement.
anonymous
  • anonymous
im talking about different question now . im asking you, what would be the growth factor if the the rate was 60% instead of 7%
anonymous
  • anonymous
according to your logic it should be 0.6 this cant be right
kropot72
  • kropot72
Are you talking about a situation where continuous annual growth is 60%. If so, assuming an initial quantity of 100 units, the quantity after 1 year would be \[N=100e ^{0.6}=182.2\ units\]
anonymous
  • anonymous
so i say, the factor should be 0.47 e^(0.47)
anonymous
  • anonymous
N=100e^0.47 = 159.99 this is much better
kropot72
  • kropot72
Your error is in expecting the amount after 1 year to be 160 units. You have not grasped the difference between discrete exponential growth and continuous exponential growth.
anonymous
  • anonymous
even in your links there is nothing that shows that your way is better
kropot72
  • kropot72
Give me details of where you found "if i look at the examples at the link you gave they do what i did"
anonymous
  • anonymous
all the examples at the first link they find the factor using my method
kropot72
  • kropot72
I do not agree that any of the examples follow your method. Your method is based on a hybrid of discrete and continuous growth formulas. Look at the explanation and examples here: http://people.stern.nyu.edu/wsilber/Continuous%20Compounding.pdf
anonymous
  • anonymous
ithink, that this approximation (which is the same as rounding) only good for low numbers
kropot72
  • kropot72
Note the following quote from the last link: " In our context, this means that if $1 is invested at 100% interest, continuously compounded, for one year, it produces $2.71828 at the end of the year. " Accord to your reasoning the amount produced should be exactly $2.00.
kropot72
  • kropot72
The formula in the link is exact, it is not an approximation.
anonymous
  • anonymous
well what tells you that any exponential grow acts like this cmpounding
anonymous
  • anonymous
the third formula at : http://mathworld.wolfram.com/CompoundInterest.html is only valid for this case i think.
anonymous
  • anonymous
also, look here : http://en.wikipedia.org/wiki/Exponential_growth#Basic_formula the way they find k
anonymous
  • anonymous
k = ln(1+r/100) / p "The percent increase r (a dimensionless number) in a period p."
anonymous
  • anonymous
now for my 60% k = 0.47 !
kropot72
  • kropot72
The following statement in the question makes it clear what formula to use: "increased exponentially at a continuous rate of 7% per year" The formula (3) in Wolfram is not appropriate. It is for discrete compounding.
kropot72
  • kropot72
The growth factor in the question is given as 7% (0.07 as a decimal). It does not have to be found!
anonymous
  • anonymous
no. look at wiki.
anonymous
  • anonymous
r = 7% and p =1 plugging in gives k = 0.067..
kropot72
  • kropot72
I am sorry you are not able to get a handle on this topic. I put it down to my limitations in explaining. I will leave this thread now. Possibly the questioner will return with information from their instructor.
anonymous
  • anonymous
cant you read what is written in wiki ?
anonymous
  • anonymous
k = ln(1+r/100) / p "The percent increase r (a dimensionless number) in a period p."
anonymous
  • anonymous
according to this, k would be 0.067 for 7% and for 60% 0.47

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