# Finding surface area with volume

#### MoneyKing

##### New member
Ok, so I did a test today for advance functions and there was a question: the volume of a pop can is 350ml, find the minimum surface area and determine the dimensions.
Where I got stuck:

350=pi(r)^2*h
h=350/pi(r)^2

SA= 2pi(r)^2+2pi(r)( h)
SA= 2pi(r)^2+2pi(r)(350/pi(r)^2)

=(2(pi(r)^3+350))/(pi(r))

I'm stuck here :S

#### Sudharaka

##### Well-known member
MHB Math Helper
Ok, so I did a test today for advance functions and there was a question: the volume of a pop can is 350ml, find the minimum surface area and determine the dimensions.
Where I got stuck:

350=pi(r)^2*h
h=350/pi(r)^2

SA= 2pi(r)^2+2pi(r)( h)
SA= 2pi(r)^2+2pi(r)(350/pi(r)^2)

=(2(pi(r)^3+350))/(pi(r))

I'm stuck here :S
Hi MoneyKing,

You should get,

$A_{s}=\frac{2(\pi r^3+350)}{r}$

Use, the first derivative test or the second derivative test to find out the value of $$r$$ which minimizes $$A_{s}$$.

Kind Regards,
Sudharaka.

#### MoneyKing

##### New member
Hi MoneyKing,

You should get,

$A_{s}=\frac{2(\pi r^3+350)}{r}$

Use, the first derivative test or the second derivative test to find out the value of $$r$$ which minimizes $$A_{s}$$.

Kind Regards,
Sudharaka.
Hey Sudharaka,
How did you get $A_{s}=\frac{2(\pi r^3+350)}{r}$
and I didn't learn how to do the first derivative test or the second, can you show me, or tell me where I can learn how to do that?

#### Sudharaka

##### Well-known member
MHB Math Helper
Hey Sudharaka,
How did you get $A_{s}=\frac{2(\pi r^3+350)}{r}$
and I didn't learn how to do the first derivative test or the second, can you show me, or tell me where I can learn how to do that?
Hi MoneyKing,

You have made a mistake in the highlighted part in post #1.

$350=\pi r^2 h\mbox{ and }A_{S}=2\pi r^2+2\pi r h$

$\Rightarrow A_{S}=2\pi r^2+2\pi r\left(\frac{350}{\pi r^2}\right)$

$\Rightarrow A_{S}=\frac{2(\pi r^3+350)}{r}$

There are loads of resources on the internet that you can learn about the first and second derivative tests. Here are two videos that you will find useful.

First Derivative Test: Calculus: 3.2 - Extrema and the First-Derivative Test - YouTube

Second Derivative Test: Calculus: Concavity and the Second-Derivative Test - YouTube

Kind Regards,
Sudharaka.

#### MoneyKing

##### New member
Hi MoneyKing,

You have made a mistake in the highlighted part in post #1.

$350=\pi r^2 h\mbox{ and }A_{S}=2\pi r^2+2\pi r h$

$\Rightarrow A_{S}=2\pi r^2+2\pi r\left(\frac{350}{\pi r^2}\right)$

$\Rightarrow A_{S}=\frac{2(\pi r^3+350)}{r}$

There are loads of resources on the internet that you can learn about the first and second derivative tests. Here are two videos that you will find useful.

First Derivative Test: Calculus: 3.2 - Extrema and the First-Derivative Test - YouTube

Second Derivative Test: Calculus: Concavity and the Second-Derivative Test - YouTube

Kind Regards,
Sudharaka.
thanks I see my mistake, umm if possible can you show me how to do this question? I still didn't take calculus, I just started Advance Functions, just finished first chapter