anonymous
  • anonymous
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)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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amistre64
  • amistre64
225 is a constant, you can pull that aside and focus on 1.15^t
amistre64
  • amistre64
ln1.15(x) = t try to substitute that
amistre64
  • amistre64
not ln...but log1.15(x) = t

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anonymous
  • anonymous
why did you take though
amistre64
  • amistre64
?
anonymous
  • anonymous
log ...im sorry
amistre64
  • amistre64
just thought itd be easier to play with.... I got no objection to another way :)
anonymous
  • anonymous
oh ok...is there a general pattern that i should follow...or even an order
amistre64
  • amistre64
i think I recall the base other than e stuff now...
amistre64
  • amistre64
a pattern..... nothing concrete that I am aware of, just find a function that you can work with easily
amistre64
  • amistre64
1.15^t is the derivative of some F(x) what do we know about exponent derivatives?
anonymous
  • anonymous
OMG im so confused
amistre64
  • amistre64
Dx(5^x) = Dx 5^x right? or am i forgetting something there...which I am prone to do
amistre64
  • amistre64
you want to integrate this function right?
anonymous
  • anonymous
yea
amistre64
  • amistre64
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?
anonymous
  • anonymous
oh ok
amistre64
  • amistre64
we dont want to change any value to it, just the way it "looks"
anonymous
  • anonymous
alright... so its log 1.15(x)= t {where did u get x from}
amistre64
  • amistre64
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"
anonymous
  • anonymous
lol..ok
amistre64
  • amistre64
lets try to "derive" a function down to something that looks close to our problem, that might help us "remember" thru the cobwebs :)
amistre64
  • amistre64
y = 5^x how would we "derive" this? do you recall?
anonymous
  • anonymous
lnx (5^x)
amistre64
  • amistre64
you sure? cause it looks right to me...
anonymous
  • anonymous
haha.. i think ...wait let me check
amistre64
  • amistre64
remember the steps to get it?
anonymous
  • anonymous
what steps?
anonymous
  • anonymous
it actually is ln(5) 5^x
amistre64
  • amistre64
thats better :)
anonymous
  • anonymous
so damn close
amistre64
  • amistre64
ok..now were ready lol
anonymous
  • anonymous
thanks for not bailing on me half way
amistre64
  • amistre64
so the derivative of 5^t would be?
amistre64
  • amistre64
i got nowhere to go ....
anonymous
  • anonymous
ln(5)5^t
amistre64
  • amistre64
and that is our "key" to this problem
amistre64
  • amistre64
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?
amistre64
  • amistre64
14 times what = 14?
anonymous
  • anonymous
*1
amistre64
  • amistre64
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?
anonymous
  • anonymous
omg...idk
amistre64
  • amistre64
lol ..... what do you know that equals 1?
amistre64
  • amistre64
does 4/2 = 1? does 4/3 =1? does 4/4 =1??
anonymous
  • anonymous
ohhh ....4/4?
amistre64
  • amistre64
instead of a "4" we need what..... ln(1.15) right?
anonymous
  • anonymous
yea
amistre64
  • amistre64
good, lets set up out integrand then...
amistre64
  • amistre64
this is our starting point \[\int\limits_{} 225(1.15^t)\]
anonymous
  • anonymous
dont we have to use F(b)-F(a)
amistre64
  • amistre64
lets pull out the constants to get this:\[225 \int\limits_{} 1.15^t\]
amistre64
  • amistre64
we need to find F(x) before we can use it lol
anonymous
  • anonymous
dang it
amistre64
  • amistre64
now lets multiply our integrand by our convenient form of "1"
amistre64
  • amistre64
\[225 \int\limits_{} [\ln(1.15)/\ln(1.15)] 1.15^t\]
amistre64
  • amistre64
tell me... is ln(1.15) a constant??
anonymous
  • anonymous
yes
amistre64
  • amistre64
then lets pull the bottom one out and leave the top in to use..sound good?
amistre64
  • amistre64
\[225/\ln(1.15) \int\limits_{} \ln(1.15) 1.15^t\]
anonymous
  • anonymous
okay that kinda makes sense
amistre64
  • amistre64
now what do we know our integrand will become?
anonymous
  • anonymous
oh shoot ......1.15^t
amistre64
  • amistre64
thats correct :) \[F(t) = [225/\ln(1.15)] 1.15^t \]
amistre64
  • amistre64
now use your "t" values and figure out the answer :)
anonymous
  • anonymous
ok...i had one more question....so my values are gonna 1 and 28?
amistre64
  • amistre64
225 ------- * 1.15^t = F(t) ln(1.15)
amistre64
  • amistre64
id subtract 1965 from both years to get a span of time
anonymous
  • anonymous
yeah i subtracted 1965 from 1983
amistre64
  • amistre64
1983?? or 1993?
anonymous
  • anonymous
1993...my bad
amistre64
  • amistre64
it aint 1 to 28, its.... zero to 28. 1965-1965=0
anonymous
  • anonymous
thats where i was confused .
amistre64
  • amistre64
1.15^0 = 1 so that aint a biggy
amistre64
  • amistre64
yeah....it was the subtraction that confused ya :) lol
anonymous
  • anonymous
OMG...i got it right...THANK YOU
amistre64
  • amistre64
80599.643 - 1609.88 =?
amistre64
  • amistre64
yay!!! that means im smart ;)
amistre64
  • amistre64
we..were smart
anonymous
  • anonymous
yea...u my friend are the man...
amistre64
  • amistre64
just dont ask me the hard questions :)
anonymous
  • anonymous
lol...its all good

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