/sci/ - Science & Math

[Anonymous No.3100836]

Hello, good people of /sci/.
I haz a math question I need help with, because my teacher is a fool. However, rather than have you answer it, I would like to be able to understand it.

If I have to make a cone with a volume of 600cm cubed, with the smallest total surface area possible, how would I set about doing this?

Any help would be much appreciated.

[ႼЋïŁႣƧტƿḩЄƦṥ !Scone2booQ No.3100849]

Do you know calculus?

[Anonymous No.3100856]

>>3100849

> Thinks this needs calculus.
> laughinggirls.png

[Anonymous No.3100858]

>>3100849
Not quite. I've been a shit student the past few years, but am really trying to turn that around.

I understand how to calculate the volume with the dimensions, but the question wants me to find the dimensions of a cone with a volume of 600cm cubed, but with the smallest possible TSA. That part I don't understand.

[Anonymous No.3100866]

>>3100858

The volume sets a definite ratio between the radius and height of your cylinder. So write what the radius must be if you are given the hight, or vice versa. Then write the surface area in terms of this, differentiate and set it equal to zero (the zero will be a minimum in this case).

For example, if the radius is related to the height by r = 4piH/16, then write out the surface area formula in terms of this, making substitutions so you have a function of one variable. This is possible because of the fixed volume.

[Scientist !!ThFjnJh4EkH No.3100868]

>>3100856
Lots of things don't "need" calculus, but that seems to be the most straightforward for someone who knows calculus. Simple optimization problem.

[Anonymous No.3100869]

First list equations, variables, and known information:

SA = pi*r*s + pi*r^2
V = (1/3)*pi*r^2*h
V = 600

Second look up the subject that you are studying:

Smallest possible anything = minima....

[ႼЋïŁႣƧტƿḩЄƦṥ !Scone2booQ No.3100883]

>>3100866
he said he didn't know calculus.

>>3100858
Like he said, the volume formula is going to set the relationship between the radius and the height. Then you're going to want to minimize the equation for surface area, based on the constraints of the volume formula. Calculus would let you solve for that directly, but once you write all that out, you might be able to eyeball it. But first put the surface area into terms of just r and h, which you should be able to do.

[Anonymous No.3100905]

He >>3100883 means s^2 = r^2 + h^2 when he says,

> But first put the surface area into terms of just r and h, which you should be able to do.

[ႼЋïŁႣƧტƿḩЄƦṥ !Scone2booQ No.3100910]

>>3100883
Once you do that, use the volume formula to get h in terms of r. Then substitute in this value where you have h in the surface area formula. Now you have the formula for the surface area of 600 volume cones of radius r, and you just have to pick the r that makes it the smallest.

Plot it on a graph, and you'll see what the situation is.

[Anonymous No.3100919]

yeah, you need calculus.
s=pi(R^4+H^2)^.5
V=600=pi(R^2)H/3
1800/Hpi=R^2
s=pi(1800^2/H^2pi^2+H^2)^.5
s'=0=pi(1800^2/H^2pi^2+H^2)^.5(2H-2*1800^2/H^3pi^2)
simplified, H=1800^2/H^3pi^2
H^4=1800^2/pi^2
H=sqrt(3*600/pi)
then use that to solve for R. so for any cone with volume X, the cone with the minimum surface area is the one where H=sqrt(3*X/pi)

[ႼЋïŁႣƧტƿḩЄƦṥ !Scone2booQ No.3100928]

>>3100919
graphing is an acceptable substitute for calculus in many situations (like this one).

[Anonymous No.3100937]

>>3100919
so you can understand it better, s' is the equation of the slope of s where H would be on the x axis. you set the slope equal to 0 because that is where either a minimum or a maximum value of S is going to be(assuming there is one).

[Anonymous No.3100944]

OK so you have the function for surface area <div class="math">f(r,h)=\pi r^{2}+\pi r \sqrt{r^{2}+h^{2}}</div>
and the constraint (volume)
<div class="math">g(r,h)=(1/3) \pi r^{2} h-600=0</div>
So we'll use a lagrange multiplier to solve it
<div class="math">L(r,h,λ)=f(r,h)-λg(r,h)</div>
And we'll set the three partial derivatives of L to zero and that just gives us three linearly independent equations for three variables and you're done.

You're welcome.

[Josef !!nUf2NflSAyw No.3100960]

>>3100944
This.
(If he hat used \lambda)

[ႼЋïŁႣƧტƿḩЄƦṥ !Scone2booQ No.3100970]

>>3100960
>>3100944
>implying a person who would ask this question would know what a lagrange multiplier is.
don't be fags.

[Anonymous No.3100991]

take equations for volume of cone and surface area of cone. set equation of volume equal to 600 and solve for one of the variables. substitute solved equation into the equation for surface area so you only have one variable. find minimum of function.

[EK FAN CLUB No.3101073]

>>3100960
>>3100944
>mfw he used a unicode lambda instead of \lambda