anonymous
  • anonymous
The mechanics at Lincoln Automotive are reboring a 6-in deep cylinder to fit a new piston. The machine they are using increases the cylinders radius one-thousandth of an inch every 4min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.800in
Mathematics
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anonymous
  • anonymous
The mechanics at Lincoln Automotive are reboring a 6-in deep cylinder to fit a new piston. The machine they are using increases the cylinders radius one-thousandth of an inch every 4min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.800in
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
okay, you can write it down like this in every 4 mins ---> radius increase = 1000in in every x mins? ----> radius increase R= D/2 =1.504 after finding the answer in mins, convert to seconds :) give it a try now :)
anonymous
  • anonymous
cross multiply
anonymous
  • anonymous
wait, I misread the question

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anonymous
  • anonymous
they want how much is the volume increasing right?
anonymous
  • anonymous
Its okay :)
anonymous
  • anonymous
at what rate is the volume increasing, supposed to use derivitives (this is calculus)
anonymous
  • anonymous
then find the derivative of Volumeof Cylinder
anonymous
  • anonymous
you have r' = 1000
anonymous
  • anonymous
all you have to do is find h from the original equation which is Vol = 1/3 pi r^2 h using r = 1.504
anonymous
  • anonymous
\[Vol = 1/3 \pi r^2h , Vol' = 2/3 \pi rr'h'\] I guess
anonymous
  • anonymous
you have r' = 1000 r= 1.504 you'll have to find h'
anonymous
  • anonymous
am I making any sense lol?
anonymous
  • anonymous
Not really...
anonymous
  • anonymous
How did you get r' ?
anonymous
  • anonymous
jmcwhs2012, if you allow, may I say a few words
anonymous
  • anonymous
it's given, it says that the radius is increasing 1000 in every 4 mins so r' = 1000in/min
anonymous
  • anonymous
iam please go ahead :)
anonymous
  • anonymous
sure:)
anonymous
  • anonymous
"one-thousandth of an inch"
anonymous
  • anonymous
Okay, as far as I understand, the rate at which the radius is increasing is equal to the radius bored in a second
anonymous
  • anonymous
you have the original radius, you can computer r' I guess it's 1000(1.504)
anonymous
  • anonymous
So we can write dr/dt= (2.54/100000)/(60*4)
anonymous
  • anonymous
compute*
anonymous
  • anonymous
Now all we need to find is dv/dt
anonymous
  • anonymous
I dont understand where you got dr/dt?
anonymous
  • anonymous
So we can use the result (dr/dt)x(dv/dr)
anonymous
  • anonymous
(dr/dt)x(dv/dr)=dv/dt
anonymous
  • anonymous
@jmcwhs2012 the thing is coming like this 2.54 cm is equal to one inch,
anonymous
  • anonymous
Why do we have to use cm?
anonymous
  • anonymous
I later convert it into m
anonymous
  • anonymous
You see for that I divide it by 100000
anonymous
  • anonymous
Ok let me explain the whole thing
anonymous
  • anonymous
The machine is working at the rate of 1/1000 inch per 60*4 secs
anonymous
  • anonymous
is that clear?
anonymous
  • anonymous
wait @iam , he wants to find V'
anonymous
  • anonymous
Thats what I am doing, if you all allow me
anonymous
  • anonymous
lol, sorry, proceed :)
anonymous
  • anonymous
The machine is working at the rate of 1/1000 inch per 60*4 secs, is that thing clear?
anonymous
  • anonymous
yes, got you.
anonymous
  • anonymous
Which means 2.54/100000 m per 240 secs
anonymous
  • anonymous
So now we can find dr/dt from there
anonymous
  • anonymous
okay
anonymous
  • anonymous
But we need to find dv/dt
anonymous
  • anonymous
so, let's find it
anonymous
  • anonymous
So we can use the formula dv/dt=(dr/dt)x(dv/dr)
anonymous
  • anonymous
Is this step clear
anonymous
  • anonymous
honestly, no, why are you multiplying them?
anonymous
  • anonymous
wait! chain rule?
anonymous
  • anonymous
Write that thing on a piece of paper, and try to cancel the commong terms, you will get why I am doing that
anonymous
  • anonymous
Yes chain rule
anonymous
  • anonymous
lol, got you! okay then what?
anonymous
  • anonymous
So we have already found dr/dt which is 2.54/100000/240
anonymous
  • anonymous
Now we need to find dv/dr
anonymous
  • anonymous
alright
anonymous
  • anonymous
For that we will have to differenciate 6 x pi x r^2 with respect to r
anonymous
  • anonymous
why 6 pi r^2?
anonymous
  • anonymous
Oops I am sorry, it would be 6*2.54*pi*r^2
anonymous
  • anonymous
6 is the length of the cylinder given on the problem
anonymous
  • anonymous
I don't get where you got 6?
anonymous
  • anonymous
oh, the height
anonymous
  • anonymous
yes
anonymous
  • anonymous
So now just we will have to multiply both the things
anonymous
  • anonymous
but isn't the vol = 1/3 pi r^2 h?
anonymous
  • anonymous
Thats for a cone
anonymous
  • anonymous
which will be 2 pi r^2?
anonymous
  • anonymous
lol...sorry ^^"
anonymous
  • anonymous
So we just need to multiply both the things
anonymous
  • anonymous
then differentiate with respect to r :)
anonymous
  • anonymous
But unfortunately the person who gave this problem is not much interested
anonymous
  • anonymous
No no no need to differenciate again
anonymous
  • anonymous
lol, he'll come back later :)
anonymous
  • anonymous
Its over
anonymous
  • anonymous
so V = 15.34 pi r^2?
anonymous
  • anonymous
yes
anonymous
  • anonymous
what is r? 1.504?
anonymous
  • anonymous
I didn't get you
anonymous
  • anonymous
r is a variable
anonymous
  • anonymous
wouldn't you derive to V' then substitute r = 1.504 and r' in the equation and then u are done?
anonymous
  • anonymous
No not 1.504, we will have to substitute 1.900
anonymous
  • anonymous
then by then you have found V' which is the volume rate
anonymous
  • anonymous
I didn't get you
anonymous
  • anonymous
alright, 1.900 and r' the one you have found which is dr/dt, then you're done
anonymous
  • anonymous
right? :)
anonymous
  • anonymous
Actually I am not getting you at all
anonymous
  • anonymous
{(2.54/100000)/240}*2.54*6*pi*2*1.900=dv/dt Thats it and thats all
anonymous
  • anonymous
so the given is: r = 1.900 r' = dr/dt = 2.54 so V' = 15.34(2) pi rr' = 30.68 pi (1.900)(2.54) = 465.15 m / secs So the volume is increasing = 465.15m/sec right?
anonymous
  • anonymous
simply ^_^
anonymous
  • anonymous
dr/dt = 2.54 this is not right
anonymous
  • anonymous
so then it's 2.54 /1000/240?
anonymous
  • anonymous
= 1.05 x10^-5
anonymous
  • anonymous
dr/dt ={(2.54/100000)/240} dv/dr= 2.54*6*pi*2*1.900
anonymous
  • anonymous
so: V' = dv/dr = 2.54*12 pi *1.900 = 181.94 m/secs

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