Question

The value, V , of a Tiffany lamp, worth $225 in 1965, increases at 15% per year. Its value in dollars t years after 1965 is given by
V=225(1.15)^t
Find the average value of the lamp over the period 1965-1993


(INTEGRAL PROBLEM)

Answer 1

225 is a constant, you can pull that aside and focus on 1.15^t

Answer 2

ln1.15(x) = t try to substitute that

Answer 3

not ln...but log1.15(x) = t

Answer 4

why did you take though

Answer 5

?

Answer 6

log ...im sorry

Answer 7

just thought itd be easier to play with.... I got no objection to another way :)

Answer 8

oh ok...is there a general pattern that i should follow...or even an order

Answer 9

i think I recall the base other than e stuff now...

Answer 10

a pattern..... nothing concrete that I am aware of, just find a function that you can work with easily

Answer 11

1.15^t is the derivative of some F(x) what do we know about exponent derivatives?

Answer 12

OMG im so confused

Answer 13

Dx(5^x) = Dx 5^x right? or am i forgetting something there...which I am prone to do

Answer 14

you want to integrate this function right?

Answer 15

yea

Answer 16

then we need to "play" around with it to get it into some form that we can work with or recognize that is easy for us right?

Answer 17

oh ok

Answer 18

we dont want to change any value to it, just the way it "looks"

Answer 19

alright... so its log 1.15(x)= t {where did u get x from}

Answer 20

from the garbage can inthe back of my memory :) i think we should try it another way... I had forgotten about the derivatives or exponential functions other then base "e"

Answer 21

lol..ok

Answer 22

lets try to "derive" a function down to something that looks close to our problem, that might help us "remember" thru the cobwebs :)

Answer 23

y = 5^x how would we "derive" this? do you recall?

Answer 24

lnx (5^x)

Answer 25

you sure? cause it looks right to me...

Answer 26

haha.. i think ...wait let me check

Answer 27

remember the steps to get it?

Answer 28

what steps?

Answer 29

it actually is ln(5) 5^x

Answer 30

thats better :)

Answer 31

so damn close

Answer 32

ok..now were ready lol

Answer 33

thanks for not bailing on me half way

Answer 34

so the derivative of 5^t would be?

Answer 35

i got nowhere to go ....

Answer 36

ln(5)5^t

Answer 37

and that is our "key" to this problem

Answer 38

if we can get it to "look" like ln(1.15) 1.15^t we can easily integrate it back up right? so how do we change the way something looks without cahnge the value of it?

Answer 39

14 times what = 14?

Answer 40

*1

Answer 41

i knew that number would be useful :) we need a convenient form of the number "1". that has ln(1.15) in it...what would that be?

Answer 42

omg...idk

Answer 43

lol ..... what do you know that equals 1?

Answer 44

does 4/2 = 1?

does 4/3 =1?

does 4/4 =1??

Answer 45

ohhh ....4/4?

Answer 46

instead of a "4" we need what..... ln(1.15) right?

Answer 47

yea

Answer 48

good, lets set up out integrand then...

Answer 49

this is our starting point
\[\int\limits_{} 225(1.15^t)\]

Answer 50

dont we have to use F(b)-F(a)

Answer 51

lets pull out the constants to get this:\[225 \int\limits_{} 1.15^t\]

Answer 52

we need to find F(x) before we can use it lol

Answer 53

dang it

Answer 54

now lets multiply our integrand by our convenient form of "1"

Answer 55

\[225 \int\limits_{} [\ln(1.15)/\ln(1.15)] 1.15^t\]

Answer 56

tell me... is ln(1.15) a constant??

Answer 57

yes

Answer 58

then lets pull the bottom one out and leave the top in to use..sound good?

Answer 59

\[225/\ln(1.15) \int\limits_{} \ln(1.15) 1.15^t\]

Answer 60

okay that kinda makes sense

Answer 61

now what do we know our integrand will become?

Answer 62

oh shoot ......1.15^t

Answer 63

thats correct :)

\[F(t) = [225/\ln(1.15)] 1.15^t \]

Answer 64

now use your "t" values and figure out the answer :)

Answer 65

ok...i had one more question....so my values are gonna 1 and 28?

Answer 66

225
------- * 1.15^t = F(t)
ln(1.15)

Answer 67

id subtract 1965 from both years to get a span of time

Answer 68

yeah i subtracted 1965 from 1983

Answer 69

1983?? or 1993?

Answer 70

1993...my bad

Answer 71

it aint 1 to 28, its.... zero to 28. 1965-1965=0

Answer 72

thats where i was confused .

Answer 73

1.15^0 = 1 so that aint a biggy

Answer 74

yeah....it was the subtraction that confused ya :) lol

Answer 75

OMG...i got it right...THANK YOU

Answer 76

80599.643 - 1609.88 =?

Answer 77

yay!!! that means im smart ;)

Answer 78

we..were smart

Answer 79

yea...u my friend are the man...

Answer 80

just dont ask me the hard questions :)

Answer 81

lol...its all good