help with optimization! (attached)

Question

anonymous · Answer

attachment

anonymous · Answer

okay i understand everything..... except how to find the minimum!

anonymous · Answer

are you familiar with derivative?

anonymous · Answer

i know they got the equation. 
y=6x+(21000/x)

anonymous · Answer

yes

anonymous · Answer

then after i'm not sure how they got:

 Minimum occurs at 59.16 ft for the length (found on a graphing calc)
width: 10500/59.16 = 177.5 ft
Perimeter = 710 ft

anonymous · Answer

ok, well the method they got the minimum is just by using a graphing calculator with a max min function on it

doing it manually would involve finding the first derivative of the equation and then setting it equal to zero and then solving for L

anonymous · Answer

or x

anonymous · Answer

is there a way to do that without it?

anonymous · Answer

without the graphing calculator?
or without having to do it manually?

anonymous · Answer

by hand

anonymous · Answer

no calculator

anonymous · Answer

by hand would involve finding the first derivative of the equation and then setting it equal to zero and then solving for L

anonymous · Answer

because the slope at the minimum point is zero

anonymous · Answer

can you show me?

anonymous · Answer

im doing it wrong, even on calculator :/ I'm not getting it.

anonymous · Answer

$$y = \frac{21000}{x} + 6x$$

$$y' = -\frac{21000}{x^2} +6$$
$$y' =0$$
$$0=-\frac{21000}{x^2} +6$$

solve for x

hopefully i didnt screw up the derivative

anonymous · Answer

$$\frac{1}{x} = x^{-1}$$

anonymous · Answer

any questions?

psymon · Answer

@completeidiot  derivative is fine :3

anonymous · Answer

yeah 1/x?

anonymous · Answer

im just pointing out the identity that allows you to use the "power" rule for derivatives

anonymous · Answer

$$y= x^n$$
$$y' = nx^{n-1}$$

anonymous · Answer

im not getting that. i got up to here:

y=6-21000/x^2 then set it to 0

anonymous · Answer

then im lost

anonymous · Answer

1/x is not part of the problem

you can ignore it if you want
its just that the identity is sometimes not obvious to other people

anonymous · Answer

once you set it equal to zero

0=6-21000/x^2

just solve for x

anonymous · Answer

6= 21000/x^2

x^2= 21000/6

x = sqrt {21000/6}

anonymous · Answer

any other questions?

sorry for any confusion i may have caused

anonymous · Answer

i got it=]

anonymous · Answer

thanks!

anonymous · Answer

no problem