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

- megannicole51

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- megannicole51

predicted which half of the century the answer would be?

- 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

im sorry i dont really understand can you elaborate

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## More answers

- anonymous

as Taplin44 said,
you need an equation:\
y(t) = Ae^(bt)
where y(0) = A (energy at 1900 which is given)

- 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

so now you can find b :
e^b = 1.07

- anonymous

sorry for my English.

- megannicole51

do u mean the exponent b or the second question?

- anonymous

Yes, he did.

- anonymous

yes i did lol

- anonymous

Coolsector is so cool!

- megannicole51

so then how would i do (c)?

- anonymous

you found the average ?

- anonymous

Yeah, did you find the average?

- anonymous

all you have to do is
average = Ae^(bt)
and now find t

- anonymous

You can do this! I believe in you1

- megannicole51

how do i bring down the exponent? its been a long night of calc and my brain is fried

- anonymous

you mean how you find
b using e^b = 1.07 ?

- megannicole51

yes

- anonymous

take ln from both sides

- anonymous

Yes, Coolsector is correct, he is never wrong!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

- megannicole51

asdfasdf54514....get a life.

- megannicole51

b=ln1.07

- anonymous

correct

- anonymous

LOL... see, he doesn't lie!!!!!!

- megannicole51

so thats the average?

- anonymous

Block me please. :) Actually never mind, I will leave.How does one untag himself?

- anonymous

no no

- anonymous

now we are only working on finding the expression for (a)!

- anonymous

im not talking about (b).
my bad for calling the parameter b

- anonymous

but it is only part of the expression

- 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

no

- megannicole51

im kinda confused

- terenzreignz

Sorry @megannicole51 I was only asking Coolsector... don't worry, you guys seem to be on the right track :D

- megannicole51

oh okay lol then keep going I'm super tired and this is my last problem! :)

- 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

so can you tell me what A is ?

- megannicole51

Y(0)?

- anonymous

y(0) = the energy consumption at the year 1900

- megannicole51

im not sure what A is then

- anonymous

y(0) = A = the energy consumption at the year 1900
it is given in the question

- megannicole51

so i was right?

- anonymous

where ?

- megannicole51

u asked what A was

- anonymous

so what is it ?

- megannicole51

idk.

- 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

so y(0) = A
and it is given in the question :
1.4 million megawatt-hours

- anonymous

so now we have:
y(t) = 1.4e^(bt)
and it satisfies our first condition :
y(0) = 1.4

- anonymous

(y in units of million megawatt-hours)

- anonymous

understand so far ?

- 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

okay go on

- anonymous

well i dont understand what is the point in giving the answers but whatever

- 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

@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

if we could keep all the numbers of b from
e^b = 1.07
we would get a growth of 7%

- anonymous

the values of y for t = 1,2,3,4.. would be accurate if we could.

- 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

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

anyway this is not really matters here.

- 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

im not confusing anything.
the way i was solving it is correct (this is what you did in fact)

- anonymous

the expression we got is continuous
but it still give the right values for every year

- kropot72

What expression did you get?

- 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

continuous expression can be still correct for specific discrete values. such as here.

- 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

my value for b is
0.067658648..
which gives
e^b = 1.07
which is the exact value

- anonymous

because you rounded b into 0.07
you got a higher precent

- anonymous

there is nothing we can do about it - we have to round.
but this is the reason.

- kropot72

No, I did not round the value. I used the decimal equivalent of 7% as given in the question.

- anonymous

ok

- 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

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

but really i dont care for it. rounding this number would give me 0.07 so i would go with it anyway

- 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

for 60% growth the factor should be about 0.47

- anonymous

please answer for my last two comments as well.

- 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

no i said instead of 7% growth
60% growth in one year.
find the factor b then.

- anonymous

according to you it should be 0.6
which cant be right

- kropot72

The growth factor remains constant when applying the equation for (a) in the question:
\[Yearly\ consumption=1.4e ^{0.07t}\]

- kropot72

You posted "according to you it should be 0.6"
I made no such statement.

- 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

according to your logic it should be 0.6
this cant be right

- 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

so i say, the factor should be 0.47
e^(0.47)

- anonymous

N=100e^0.47 = 159.99
this is much better

- 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

even in your links there is nothing that shows that your way is better

- 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

all the examples at the first link
they find the factor using my method

- 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

ithink, that this approximation (which is the same as rounding) only good for low numbers

- 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

The formula in the link is exact, it is not an approximation.

- anonymous

well what tells you that any exponential grow acts like this cmpounding

- anonymous

the third formula at :
http://mathworld.wolfram.com/CompoundInterest.html
is only valid for this case i think.

- anonymous

also, look here :
http://en.wikipedia.org/wiki/Exponential_growth#Basic_formula
the way they find k

- anonymous

k = ln(1+r/100) / p
"The percent increase r (a dimensionless number) in a period p."

- anonymous

now for my 60%
k = 0.47 !

- 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

The growth factor in the question is given as 7% (0.07 as a decimal). It does not have to be found!

- anonymous

no. look at wiki.

- anonymous

r = 7% and p =1
plugging in gives
k = 0.067..

- 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

cant you read what is written in wiki ?

- anonymous

k = ln(1+r/100) / p
"The percent increase r (a dimensionless number) in a period p."

- anonymous

according to this,
k would be 0.067 for 7%
and for 60% 0.47

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