Question

Radius and height for smallest surface area given volume of an open cone is 27cm^3?

Answer 1

what is the surface area of a cone equaton

Answer 2

For an open cone:
\[SA = \pi r h\]

Answer 3

okay so lets optimize

Answer 4

what isthe volume of a cone formula?

Answer 5

Volume:
27=13 basearea(r2)

Base Area:
A=πr2

Base area into Volume:
27=13πr2h

So:
h=81πr2

Putting that into Surface Area:
SA=πr81πr2

=81r

Answer 6

But taking the derivative of SA:
SA′=−81r2

And since SA is at a min/max when SA' = 0:
r2=−81

And then:
r=9i

Which doesn't work in the question.

Answer 7

that simplification is wrong

Answer 8

Sorry, 13s should be 1/3, and the ^2 didn't work.

Answer 9

Let me try again...

Answer 10

81 pi^2 * r^3

Answer 11

Volume:
\[27=\frac{ 1 }{ 3 } basearea(r^2)\]

Base Area:
\[A=πr^2\]

Base area into Volume:
\[27=\frac{ 1 }{ 3 }πr^2h\]

So:
\[h=\frac{ 81 }{ πr^2 }\]

Putting that into Surface Area:
\[SA=\frac{ πr81 }{ πr^2 }\]

\[=\frac{ 81 }{ r }\]

Answer 12

ok

Answer 13

So far so good?

Answer 14

But taking the derivative of SA:
\[SA′=\frac{ -81 }{ r^2 }\]

And since SA is at a min/max when SA' = 0:
\[r^2=−81\]

And then:
\[r=9i\]

Which doesn't work in the question.

Answer 15

tha tsimplficiation is not right

Answer 16

u will get 1/r = 0 instead but even then

Answer 17

At which step?

Answer 18

-81/r^2 = 0

Answer 19

dividing both sides and sqrting both side gives u 1/r = 0

Answer 20

okay, so what should I do?

Answer 21

well this still isnt giving u very nice answers let me try doing it from start see if there is some problem

Answer 22

because right now if we solve this.. r= +/- inf and h will be going to 0 which doesnt make sense

Answer 23

i see ur problem

Answer 24

the SA =pi r*(sqrt(h^2+r^2))

Answer 25

But that's for a cone including the base, isn't it? The question asks for a "cone shaped paper drinking cup" so shouldn't it be open? or does this not matter?

Answer 26

oh u are right

Answer 27

Okay, no, that is the equation for an open cone. Right, I'll try again...

Answer 28

oh okay yeah

Answer 29

becayse it SA= pi*r*L

Answer 30

L=sqrth^2+r^2

Answer 31

Yep :)

Answer 32

Surface Area (given open top):
\[SA=πr \sqrt{h^2+r^2} \]

Volume:
\[27=\frac{ 1 }{ 3 } basearea(r^2)\]

Base Area:
\[A=πr^2\]

Base area into Volume:
\[27=\frac{ 1 }{ 3 }πr^2h\]

So:
\[h=\frac{ 81 }{ πr^2 }\]

Putting that into Surface Area:
\[SA=πr \sqrt{\frac{ 81 }{ \pi r^2 }+r^2}\]

Answer 33

ok

Answer 34

bring the pi r into sqrt and simplify

Answer 35

\[SA = \sqrt{(\frac{ 81 }{ \pi r^2 }+r^2)\pi^2 r^2}\]
Is that right?

Answer 36

yep simplify that now

Answer 37

d/dr sqrt(pi 81 +pi^2 r^3) = 3pi^2r^2/2 (pi 81 +pi^2 r^3)^(-1/2)

Answer 38

Shouldn't it be d/dr sqrt(pi 81 + pi^2 r^((4)))?
(changed 3 to 4 in extra brackets)

Answer 39

So:
\[SA = \sqrt{81\pi + r^4 \pi^2}\]
\[SA' = \frac{ 1 }{ 2 }(81\pi + r^4 \pi^2)^{-1/2}(4r^3 \pi^2)\]

Answer 40

But then SA' = 0 still gives r=0

Answer 41

What went wrong?

Answer 42

what formula did u use for SA?

Answer 43

See 3rd post down
PI*r*h

Answer 44

that's incorrect

Answer 45

No, changed later to SA = pi r sqrt(h^2 + r^2)

Answer 46

Sorry wiggler

Answer 47

Haha, all good :) It's a LONG work in progress

Answer 48

did u substitute h or r above with the volume?

Answer 49

substituted in h = 81/r^2 to the SA eqn

Answer 50

(eliminated h in order to find r)

Answer 51

u missing the pi in ur substiute, u sure of it?

Answer 52

well gotta go - seems kx2bay will be a BIG help.
see you folks
:-)

Answer 53

Where is the pi missing?

Answer 54

u said above "substituted in h = 81/r^2 to the SA eqn", where is the pi ?
volume = 27 = 1/3 pi r2 h

Answer 55

Oh, yeah, I did include that in the calculations.

Answer 56

Copying error

Answer 57

That gave me SA = pi r * sqrt((81/pi r^2) + r^2)

Answer 58

I started with subst. r = 9/sqrt(pi.h)
and got at the end this
\[9 (81.h ^{-2} + \pi.h)^{\frac{ 1 }{ 2 }}\]

this is where am struggling to find the derivative, do u know what it will be?

Answer 59

\[4.5(81h^-2 + \pi h)(-162h^-3 + \pi)\]

Answer 60

That should be h^-2 and h^-3

Answer 61

thanks
first answer for h is negative i.e. h = cubeRoot(-pi/81)

Answer 62

second answer for h is good i.e.
h^3 = 162/pi

Answer 63

so h = 3.722 cm
what is r ? based on
r=9/sqrt(pi.h)

Answer 64

i got r = 2.632 cm
lets check the volume now if it works

Answer 65

yep I get 27 for the volume
so that's ur answers
h = 3.722
and r = 2.632

now teach me how did you work out the derivative above that got stuck on because I need to refresh my rusty memory on that :) please...

Answer 66

that I got stuck on

Answer 67

Sure :) You start with the outer function (the power of 1/2)
Taking the stuff in the brackets as "x", take the derivative of 9x^(1/2)

Answer 68

ok, continue please

Answer 69

it will give u 1/2 *9*x^-1/2

Answer 70

Yep. Then multiply that by the derivative of the inner term.
i.e. what you just calculated * derivative of (81/h^2 + pi h)

Answer 71

hmmm u too good with stuff, thanks for the tutoring :)

Answer 72

No problem :) just remember to start with the outermost function and work your way in. And thanks for your help!

Answer 73

no worries and thank you
cheers