What is the derivative of the equation for volume?

[stuckie27]

Smiple Equation.

What is the derative of 3pi*r^2+ ((2,000,000-(2/3pi*r^3))/r)

Do you use the quotient rule

 

[cookiemonster]

It'd work, but it's more work than you need to do.

That same expression can be written as

$$\frac{7}{3}\pi r^2 + 2\times 10^6 r^{-1}$$

which can be differentiated easily.

cookiemonster

 

[JonF]

Yup you use the quotient rule. This is what I got when I did (warning, I am prone to make arithmetic mistakes)

$6\pir + \frac{ r (\frac{18/pir^2}{3/pir^5}) –(2,000,000 – (\frac{2}{3\pir^3}){r^2}$

 

[cookiemonster]

(warning, I am prone to make arithmetic mistakes)

I agree, which is why you should simplify the expression before differentiating it.

cookiemonster

 

[stuckie27]

Thanks for the quick reply.

3 cookies for cookiemonster (extra choclate chip)

how did you simplify the equation?

 

[Ebolamonk3y]

sigh... waste of good time on these trivial problems...

 

[cookiemonster]

Distributed the r^(-1) in the second term and then combined coefficients of like powers of r.

And a trivial problem to you, Ebolamonk3y, may not be trivial to somebody else.

cookiemonster

 

[stuckie27]

6\pir + \frac{ r (\frac{18/pir^2}{3/pir^5}) –(2,000,000 – (\frac{2}{3\pir^3}){r^2}


What is this? \frac{ r (\frac{18/pir^2}{3/pir^5}) Its not in my original equation

 

[Ebolamonk3y]

use $$...$$

 

[stuckie27]

sigh... waste of good time on these trivial problems...

Any reason for this comment?

Is this not a calc help forum?

I can write you the full prolbem if you would like to solve the whole thing.

Here It is

A metal storage tank with a volume of 1000 liters is to be constructed in the shape of a right cylinder surmounted by a hemisphere. What dimentions will require the least amount of metal?

 

[cookiemonster]

It's supposed to be the expression for the derivative of your original expression. I think. I can't read it well enough to really know, though.

cookiemonster

 

[cookiemonster]

Let's not turn this into a flame war, okay?

cookiemonster

 

[Ebolamonk3y]

Well... long time ago I proposed this huge problem that I made up for a friend of mine... by the end of it I noticed that the problem didn't led to anywhere but a handful of arithemetic exercises... And that doesn't serve any purpose, I am not udnersetanding more just because I did some rote problems... It was some derivative involving the chainrule and this stuff where there is so many layers and functions of functions that one get lost it in and finally one wonders, what is this for? Waste of time for me... So that's why I said that... Take no offense, that is how I felt about my experience...

Like... if one wants to really find antiderivated to e^(x^2)... because one didn't read about it, one this they can use some stuff on it and then they realize... :(

Sorry about that.

 

[JonF]

[sarcasm] But… but… my brain can beat up your brain [/sarcasm]

 

[Ebolamonk3y]

heh. perhaps. my mind is afflicted with many things like depression to easily succumb to an incoming invasion of anothers mind. :p

 

[Integral]

Lets start from scratch.
The volume is the sum of the volume of a hemisphere and a cylinder.
$$V= \frac 2 3 \pi r^3 + \pi r^2 h$$
The surface area is
$$S= 2 \pi r h + 2 \pi r^2$$
Isolate h in the Volumn equation.
$$h = \frac {V- \frac 2 3 \pi r^2} {\pi r^2}$$
plug into the Surface area equation.
$$S= 2 \pi r \frac {V- \frac 2 3 \pi r^2} {\pi r^2} + 2 \pi r^2$$

$$S= \frac {2 V} r + \frac 2 3 \pi r^2$$
Compute the derivative.
$$\frac {dS} {dr} = - \frac {2V} {r^2} + \frac 4 3 \pi r = 0$$
Solve for r
$$-2V + \frac 4 3 \pi r^3 = 0$$

$$r^3 = \frac {3V} {2 \pi}$$

Complete by solving for h.