Question

Approximate the volume of a spherical shell where the inner radius is 4 inches and whose thickness is 1/16 inches using DIFFERENTIALS.

Answer 1

hi. do you still need help with this problem?

Answer 2

yes. will you help me?

Answer 3

i will do what i can, yes

Answer 4

so let's start off with whether or not we can find some applicable formulas here.

Answer 5

your heaven sent thank you..:)

Answer 6

do you have the answer to this thing?

Answer 7

none..

Answer 8

k. Let's start with what we know. Like maybe the volume of a sphere.

Answer 9

Do you remember the formula for the volume of a sphere? dont worry, you are gonna get through this problem. I promise

Answer 10

(pi)(d)^3/6

Answer 11

or 4/3 (pi) R^3

Answer 12

yes... that is what we seek..

Answer 13

now what is the variable that affects the volume? Because that is the variable that will drive the differentiation process.

Answer 14

Radius?

Answer 15

absolutely.. good. Now how do we get the differential of this little puppy?

Answer 16

solve for V' ?

Answer 17

what is it we are approximating?

Answer 18

wait. i didnt see the prime

Answer 19

the volume of a spherical shell

Answer 20

yes. we want V'

Answer 21

so how does one go about getting V'?

Answer 22

in reality, it is dV/dr we need.

Answer 23

\[\frac{ dv }{ dr }=(\frac{ 4 }{ 3 })(\pi)(3R)\]
not sure..is it right?

Answer 24

you are very close. check the exponent on the R term

Answer 25

3R^2 :)

Answer 26

yes. so the whole differential looks like.... dV=

Answer 27

remember that dV=(dV/dr)dr

Answer 28

dv = (4/3)(pi)(3R^2)dr ?

Answer 29

perfect!

Answer 30

notice how the 3's cancel

Answer 31

for somebody who desperately thought you could nt do this problem, you are doing remarkably well :)

Answer 32

dv = 4 pi R^2 dr

Answer 33

yes. now plug in your given values and u are done. excellent work

Answer 34

notice how you can approximate the change in volume by taking the product of the surface area of the sphere and dr.

Answer 35

where do i substitute 4 inches to? 1/16 inches?

Answer 36

r=4 and dr=1/16

Answer 37

dv = 12.5663... ?

Answer 38

this approximation is good because the dr or differential of r is very small ( 1/16)

Answer 39

ummm... i dont have a calc with me. but that looks pretty close to 4pi, which is the answer i got

Answer 40

thank you (1000000000x)!!!

Answer 41

it was such a pleasure

Answer 42

wait one last question...

Answer 43

sure. shoot.

Answer 44

find the value of cos 43 using differentials...

Answer 45

lol. i think i saw that problem an hour or two ago.

Answer 46

yes..but no one helped me :(

Answer 47

i am so sorry. i saw someone looking so i decided to move on.

Answer 48

ok. the drill is the same....except... ummm... slightly different. and no, that is not a DIFFERENTial joke.

Answer 49

so where do you think you might start? what is the reference angle you would use in this case? 43 degrees is very close to .....

Answer 50

45 degrees

Answer 51

good. that is exactly correct

Answer 52

what function are we differentiating?

Answer 53

cos

Answer 54

y=....

Answer 55

nice

Answer 56

now i could be mistaken, but i think we need to do this in radians

Answer 57

so? what's the next step?

Answer 58

find the formula for the differential given that y=cos x

Answer 59

dy/dx = -sin x ?

Answer 60

perfect. and so dy=?

Answer 61

-sin x dx

Answer 62

yes!

Answer 63

x=? dx?

Answer 64

no, dy=-sin x dx

Answer 65

i mean
what is x=?
dx=?

Answer 66

x=45=pi/4 and dx=-2=-pi/90

Answer 67

why -pi/90?

Answer 68

cuz that is 2 degrees in radians. all trig function are calculated in radians in calc. cant remember for the life of me why tho. sorry about that

Answer 69

(-2)xpi/180=-pi/90. the pi/180 converts degrees to radians

Answer 70

remember that dx=delta x=43-45=-2

Answer 71

ahh..gets

Answer 72

if you are not comfy cozy with this, please feel free to ask me questions

Answer 73

is dx= change in x?

Answer 74

yes, by the definition of a differential. the differential of y, dy, equals y'x dx, where dx=the change in x

Answer 75

oops.. the differential of y still equals y'x dx, not y'x dx.

Answer 76

hmmm

Answer 77

y'dx

Answer 78

there we go

Answer 79

oh okay...thanks a lot again!!
i hope you'll never get tired of helping me..
see ya next time??

Answer 80

i hope so. it has been a pleasure. you take care unk. lol. bye