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

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- anonymous

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- 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

cross multiply

- anonymous

wait, I misread the question

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## More answers

- anonymous

they want how much is the volume increasing right?

- anonymous

Its okay :)

- anonymous

at what rate is the volume increasing, supposed to use derivitives (this is calculus)

- anonymous

then find the derivative of Volumeof Cylinder

- anonymous

you have r' = 1000

- 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

\[Vol = 1/3 \pi r^2h , Vol' = 2/3 \pi rr'h'\] I guess

- anonymous

you have
r' = 1000
r= 1.504
you'll have to find h'

- anonymous

am I making any sense lol?

- anonymous

Not really...

- anonymous

How did you get r' ?

- anonymous

jmcwhs2012, if you allow, may I say a few words

- anonymous

it's given, it says that the radius is increasing 1000 in every 4 mins so r' = 1000in/min

- anonymous

iam please go ahead :)

- anonymous

sure:)

- anonymous

"one-thousandth of an inch"

- 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

you have the original radius, you can computer r' I guess it's 1000(1.504)

- anonymous

So we can write dr/dt= (2.54/100000)/(60*4)

- anonymous

compute*

- anonymous

Now all we need to find is dv/dt

- anonymous

I dont understand where you got dr/dt?

- anonymous

So we can use the result
(dr/dt)x(dv/dr)

- anonymous

(dr/dt)x(dv/dr)=dv/dt

- anonymous

@jmcwhs2012 the thing is coming like this 2.54 cm is equal to one inch,

- anonymous

Why do we have to use cm?

- anonymous

I later convert it into m

- anonymous

You see for that I divide it by 100000

- anonymous

Ok let me explain the whole thing

- anonymous

The machine is working at the rate of 1/1000 inch per 60*4 secs

- anonymous

is that clear?

- anonymous

wait @iam , he wants to find V'

- anonymous

Thats what I am doing, if you all allow me

- anonymous

lol, sorry, proceed :)

- anonymous

The machine is working at the rate of 1/1000 inch per 60*4 secs, is that thing clear?

- anonymous

yes, got you.

- anonymous

Which means 2.54/100000 m per 240 secs

- anonymous

So now we can find dr/dt from there

- anonymous

okay

- anonymous

But we need to find dv/dt

- anonymous

so, let's find it

- anonymous

So we can use the formula dv/dt=(dr/dt)x(dv/dr)

- anonymous

Is this step clear

- anonymous

honestly, no, why are you multiplying them?

- anonymous

wait! chain rule?

- 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

Yes chain rule

- anonymous

lol, got you! okay then what?

- anonymous

So we have already found dr/dt which is 2.54/100000/240

- anonymous

Now we need to find dv/dr

- anonymous

alright

- anonymous

For that we will have to differenciate 6 x pi x r^2 with respect to r

- anonymous

why 6 pi r^2?

- anonymous

Oops I am sorry, it would be 6*2.54*pi*r^2

- anonymous

6 is the length of the cylinder given on the problem

- anonymous

I don't get where you got 6?

- anonymous

oh, the height

- anonymous

yes

- anonymous

So now just we will have to multiply both the things

- anonymous

but isn't the vol = 1/3 pi r^2 h?

- anonymous

Thats for a cone

- anonymous

which will be 2 pi r^2?

- anonymous

lol...sorry ^^"

- anonymous

So we just need to multiply both the things

- anonymous

then differentiate with respect to r :)

- anonymous

But unfortunately the person who gave this problem is not much interested

- anonymous

No no no need to differenciate again

- anonymous

lol, he'll come back later :)

- anonymous

Its over

- anonymous

so V = 15.34 pi r^2?

- anonymous

yes

- anonymous

what is r? 1.504?

- anonymous

I didn't get you

- anonymous

r is a variable

- anonymous

wouldn't you derive to V' then substitute r = 1.504 and r' in the equation and then u are done?

- anonymous

No not 1.504, we will have to substitute 1.900

- anonymous

then by then you have found V' which is the volume rate

- anonymous

I didn't get you

- anonymous

alright, 1.900 and r' the one you have found which is dr/dt, then you're done

- anonymous

right? :)

- anonymous

Actually I am not getting you at all

- anonymous

{(2.54/100000)/240}*2.54*6*pi*2*1.900=dv/dt
Thats it and thats all

- 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

simply ^_^

- anonymous

dr/dt = 2.54 this is not right

- anonymous

so then it's 2.54 /1000/240?

- anonymous

= 1.05 x10^-5

- anonymous

dr/dt ={(2.54/100000)/240}
dv/dr= 2.54*6*pi*2*1.900

- anonymous

so:
V' = dv/dr = 2.54*12 pi *1.900
= 181.94 m/secs

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