Question

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

Answer 1

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 :)

Answer 2

cross multiply

Answer 3

wait, I misread the question

Answer 4

they want how much is the volume increasing right?

Answer 5

Its okay :)

Answer 6

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

Answer 7

then find the derivative of Volumeof Cylinder

Answer 8

you have r' = 1000

Answer 9

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

Answer 10

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

Answer 11

you have

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

Answer 12

am I making any sense lol?

Answer 13

Not really...

Answer 14

How did you get r' ?

Answer 15

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

Answer 16

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

Answer 17

iam please go ahead :)

Answer 18

sure:)

Answer 19

"one-thousandth of an inch"

Answer 20

Okay, as far as I understand, the rate at which the radius is increasing is equal to the radius bored in a second

Answer 21

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

Answer 22

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

Answer 23

compute*

Answer 24

Now all we need to find is dv/dt

Answer 25

I dont understand where you got dr/dt?

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

Why do we have to use cm?

Answer 30

I later convert it into m

Answer 31

You see for that I divide it by 100000

Answer 32

Ok let me explain the whole thing

Answer 33

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

Answer 34

is that clear?

Answer 35

wait @iam , he wants to find V'

Answer 36

Thats what I am doing, if you all allow me

Answer 37

lol, sorry, proceed :)

Answer 38

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

Answer 39

yes, got you.

Answer 40

Which means 2.54/100000 m per 240 secs

Answer 41

So now we can find dr/dt from there

Answer 42

okay

Answer 43

But we need to find dv/dt

Answer 44

so, let's find it

Answer 45

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

Answer 46

Is this step clear

Answer 47

honestly, no, why are you multiplying them?

Answer 48

wait! chain rule?

Answer 49

Write that thing on a piece of paper, and try to cancel the commong terms, you will get why I am doing that

Answer 50

Yes chain rule

Answer 51

lol, got you! okay then what?

Answer 52

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

Answer 53

Now we need to find dv/dr

Answer 54

alright

Answer 55

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

Answer 56

why 6 pi r^2?

Answer 57

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

Answer 58

6 is the length of the cylinder given on the problem

Answer 59

I don't get where you got 6?

Answer 60

oh, the height

Answer 61

yes

Answer 62

So now just we will have to multiply both the things

Answer 63

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

Answer 64

Thats for a cone

Answer 65

which will be 2 pi r^2?

Answer 66

lol...sorry ^^"

Answer 67

So we just need to multiply both the things

Answer 68

then differentiate with respect to r :)

Answer 69

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

Answer 70

No no no need to differenciate again

Answer 71

lol, he'll come back later :)

Answer 72

Its over

Answer 73

so V = 15.34 pi r^2?

Answer 74

yes

Answer 75

what is r? 1.504?

Answer 76

I didn't get you

Answer 77

r is a variable

Answer 78

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

Answer 79

No not 1.504, we will have to substitute 1.900

Answer 80

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

Answer 81

I didn't get you

Answer 82

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

Answer 83

right? :)

Answer 84

Actually I am not getting you at all

Answer 85

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


Thats it and thats all

Answer 86

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?

Answer 87

simply ^_^

Answer 88

dr/dt = 2.54 this is not right

Answer 89

so then it's 2.54 /1000/240?

Answer 90

= 1.05 x10^-5

Answer 91

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

Answer 92

so:

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