Question

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

Answer 1

predicted which half of the century the answer would be?

Answer 2

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)

Answer 3

im sorry i dont really understand can you elaborate

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

sorry for my English.

Answer 8

do u mean the exponent b or the second question?

Answer 9

Yes, he did.

Answer 10

yes i did lol

Answer 11

Coolsector is so cool!

Answer 12

so then how would i do (c)?

Answer 13

you found the average ?

Answer 14

Yeah, did you find the average?

Answer 15

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

Answer 16

You can do this! I believe in you1

Answer 17

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

Answer 18

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

Answer 19

yes

Answer 20

take ln from both sides

Answer 21

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

Answer 22

asdfasdf54514....get a life.

Answer 23

b=ln1.07

Answer 24

correct

Answer 25

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

Answer 26

so thats the average?

Answer 27

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

Answer 28

no no

Answer 29

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

Answer 30

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

Answer 31

but it is only part of the expression

Answer 32

@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 \[\Large A(1.07)^t\]?

Answer 33

no

Answer 34

im kinda confused

Answer 35

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

Answer 36

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

Answer 37

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!

Answer 38

so can you tell me what A is ?

Answer 39

Y(0)?

Answer 40

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

Answer 41

im not sure what A is then

Answer 42

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

Answer 43

so i was right?

Answer 44

where ?

Answer 45

u asked what A was

Answer 46

so what is it ?

Answer 47

idk.

Answer 48

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)

Answer 49

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

Answer 50

so now we have:
y(t) = 1.4e^(bt)

and it satisfies our first condition :
y(0) = 1.4

Answer 51

(y in units of million megawatt-hours)

Answer 52

understand so far ?

Answer 53

The equation for (a) is:
\[Yearly\ consumption=1.4e ^{0.07t}\]
(b)
\[Average\ thru \ 20th\ century=\frac{\int\limits_{t=0}^{t=100}1.4e ^{0.07t}.dt}{100}\]

Answer 54

okay go on

Answer 55

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

Answer 56

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)?

Answer 57

@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.

Answer 58

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

Answer 59

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

Answer 60

@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

Answer 61

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.

Answer 62

anyway this is not really matters here.

Answer 63

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

Answer 64

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

Answer 65

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

Answer 66

What expression did you get?

Answer 67

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"

Answer 68

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

Answer 69

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.

Answer 70

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

Answer 71

because you rounded b into 0.07
you got a higher precent

Answer 72

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

Answer 73

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

Answer 74

ok

Answer 75

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
\[A=100e ^{0.07}=$107.251\]

Answer 76

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

Answer 77

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

Answer 78

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

Answer 79

for 60% growth the factor should be about 0.47

Answer 80

please answer for my last two comments as well.

Answer 81

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.

Answer 82

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

Answer 83

according to you it should be 0.6
which cant be right

Answer 84

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

Answer 85

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

Answer 86

im talking about different question now .
im asking you, what would be the growth factor
if the the rate was 60% instead of 7%

Answer 87

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

Answer 88

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
\[N=100e ^{0.6}=182.2\ units\]

Answer 89

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

Answer 90

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

Answer 91

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.

Answer 92

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

Answer 93

Give me details of where you found "if i look at the examples at the link you gave they do what i did"

Answer 94

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

Answer 95

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

Answer 96

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

Answer 97

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.