Question

Im given the volume and surface area formulas:
The accompanying figure gives formulas for the volume (V), and the total surface area (S) of a circle Cylindar with a radius (r) and height (h).
a) Assume the volume of the cylinder is 1000 cm^3. Express the surface area as a function of the radius. After Combining terms in your answer, show that the resulting function can be written as:
S(r)= 2pi*r^2 + 2000 / r (r>0)

b) Use a graphing calculator to graph the surface area function in part ac) Estimate to the nearest one-hundredth, the radius r that minimizes the surface area. What is the..

Answer 1

I know its not a very good photo. but Ill type out everything that it says:::

Answer 2

I dont know where, or how to start. its number 38) Has 3 questions a,b &c

The accompanying figure gives formulas for the volume (V), and the total surface area (S) of a circular cylindar with a radius r and height h.

\[V=\pi r^2 h\]
\[S=2 \pi r^2 + 2 \pi r^2 h\]

a) assume the volume of the Cylindar is \[1000cm ^{3}\] . Express the surface area as a function of the radius. After combinging terms in your answer whow that the resulting function can be written: \[S _{r} = \frac{ 2 \pi r^3 +2000 }{ r }\]

b) Use a graphing utility to graph the Surface Area function in part (a)

c) Estimate to the nearest one-hundredth, the radiu r that minimizes the surface area. What is the corresponding value for h?

Answer 3

Ok ok ok need help with a) ?

Answer 4

I actually need....ya I guess a will help me with the rest of the questions right?

Yes please @zepdrix , could you help me with a) ??

Answer 5

\[\Large V=\pi r^2h \qquad\to\qquad 1000=\pi r^2 h\]Solving for h gives us,\[\Large \color{#CC0033}{h=\frac{1000}{\pi r^2}}\]

Answer 6

We want to use this relationship between h and r and plug it into our surface area formula.

Answer 7

\[\Large S=2 \pi r^2 + 2 \pi r^2 \color{#CC0033}{h}\]

Answer 8

oh okay, Hold on I think i see what you did. u plugged in the 1000 that was volume. and like you said we solved for h that gave us that equation.

Answer 9

Oh your formula is a little funky, let's fix that real quick.
The Surface area formula shouldn't have an r^2 on the second term.\[\Large S=2 \pi r^2 + 2 \pi r \color{#CC0033}{h}\]

Answer 10

okay then so it would look like this: then \[S= 2 \pi r^2 + 2 \pi r^2 (\frac{ 1000 }{ \pi r^2 })\] right?

Answer 11

yup looks good!

Answer 12

okay umm.. i have a questions.

Answer 13

can the pi r^2 on the bottom of the 1000 be crossed out with the one right by it giving me .....this 2(1000) or am I wrong? @zepdrix

Answer 14

Look at my last formula I posted D: There should NOT be an r^2 on that second term.
So they wouldn't completely cancel out like that.

Answer 15

okay let me look

Answer 16

OH! haha oops sorry. haha okay thank you.
But can I still reduce that pi r^2 ?

Answer 17

Ya it looks the pi's will cancel out right?
And then we can cancel out the r with ONE OF the r's in the bottom.

Answer 18

okay let me see. i get......

Answer 19

\[S= 2 \pi r^2 + 2 \left(\begin{matrix}1000 \\ r\end{matrix}\right)\]

Answer 20

is this right @zepdrix ? it looks odd to me

Answer 21

that's supposed to be a fraction, right? lol
ya looks good so far!

From there, let's combine our two terms by getting a common denominator.
Oh first multiply the 2 and 1000 together.

Answer 22

\[\Large S=2\pi r^2+\frac{2000}{r}\]

Answer 23

why doesnt the 2 multiply with r?

Answer 24

Multiplying by a fraction:
\[\large 2\times\frac{3}{4} \quad=\quad \frac{2}{1}\times\frac{3}{4} \quad=\quad \frac{2\times3}{1\times4}\]

Answer 25

Understand how that works? :o
The 2 is `technically` in the numerator, even though didn't write a denominator under it.
So we multiply the numerators together.

Answer 26

oh haha okay now i see it. i feel dumb now. haha sorry about that. haha

Answer 27

Okay so I end up getting \[S = 2 \pi r^2 + \frac{ 2000 }{ r }\]

Answer 28

From there we need to combine the two terms.
So we need a common denominator.

Answer 29

okay then it gives me...............just a sec

Answer 30

\[2 \pi r^3 + 2000r\]

Answer 31

over r? :o

Answer 32

@zepdrix is this the right formula?

Answer 33

mm im not sure what you did +_+

Answer 34

To get a common denominator, we multiply the first term by r/r.\[\Large S=2\pi r^2\cdot\color{#3366CF}{\frac{r}{r}}+\frac{2000}{r}\]

Answer 35

Which gives us,\[\Large S=\frac{2\pi r^3}{r}+\frac{2000}{r}\]
See how they have the same denominator now? :o

Answer 36

This allows us to write them as a single fraction.\[\Large S=\frac{2\pi r^3+2000}{r}\]

Answer 37

oh...............okay thanks, I think i was multiplying (r/1) to each side haha

Answer 38

thank you @zepdrix okay well now for b) i put it in my graphing calucator. well I am right now.

Answer 39

HAHA i dont know how to plug it into a graphing calcualtor.

Answer 40

hmm

Answer 41

This is a really nice graphing utility that I like to use.
https://www.desmos.com/calculator

Answer 42

ya, i cant get it to work on my calculator. is there a way to do it, or do i just graph?

Answer 43

You would want to graph it in x,y. Not in S,r.
So make sure of that first of all :D

Answer 44

lol ya i have it under x y

Answer 45

On your calculator I guess you would want to input something like:\[\Large y=(2\pi r\wedge3+2000)/r\]

Answer 46

AHH i did r's lol.

Answer 47

\[\Large y=(2\pi x\wedge3+2000)/x\]

Answer 48

haha

Answer 49

This graph is really really difficult to read though.. hmm

Answer 50

haha i keep getting error

Answer 51

?

Answer 52

YAY!!!! now i got a freakin line haha ://

Answer 53

It's not a line. It's just really hard to see the curve.
It might be a little easier to read if you look at it here.
https://www.desmos.com/calculator/lx6c49dxwt

Answer 54

Can you see how the curve dips down and then back up somewhere around y=600?

Answer 55

Ya I do see it

Answer 56

okay Ill keep that site up, but for c it now says.... to estimate to the nearest one hundredth, the radius r that minimizes the surface area. What is the corresponding value for h?

Answer 57

So we need to find an r ( remember, r=x on our graph ) that minimizes the surface area.

Then we'll use that r value in here:\[\Large \color{#CC0033}{h=\frac{1000}{\pi r^2}}\]To find a corresponding h value.

Answer 58

Minimize umm we'll just simplify that for now to mean ~ where does the graph dip down?

Answer 59

If you hold your mouse down on the curve, you can move across it and it'll show you the coordinates.
Try to get the point at the bottom of that dip.

Answer 60

( 5.4 , 553.6 )
Were you able to find that point?

Answer 61

okay ill try it

Answer 62

looks about....................(5 2/3 , 553 2/3)

Answer 63

It should give you an actual point at the bottom of the curve, right? :o
A black dot.

Answer 64

okay it says. : 5.419, 553.581

Answer 65

Ok good.
That is written as a coordinate pair ( r , S ) = ( 5.419 , 553.581 )

So our r value, rounded to the nearest hundredth, will be r = 5.4. Right? :)

Answer 66

Woops I rounded to the tenths :P derp!

Answer 67

r=5.42 I guess

Answer 68

haha okay got 5.42

Answer 69

Ok good, we can find our corresponding h value now:\[\Large \color{#CC0033}{h=\frac{1000}{\pi (5.42)^2}}\]

Answer 70

do i go ahead then since we found r haha ya I plugged it in

Answer 71

it gives me 10.83

Answer 72

is that what you got?

Answer 73

mmmm yah that's what im getting also

Answer 74

So it's probably right :3 lol

yay team \c:/

Answer 75

HAHA YAY!!!! TEAM!!! HIGH FIVE!!! ^ . ^ (V)

Answer 76

@zepdrix thanks so much, this has been driving me nuts for 2 days now haha

Answer 77

hehe np :o