Question

Solve the following Optimization problem. Must justify results. Decimal answers are acceptable.

A square and/or a rectangle are formed using 50 inches of wire. The rectangle has a base to height ratio of 3 to 1. Determine the minimum total area.

Answer 1

P = 50
P = 4x + 8y
50 = 4x + 8y
y = (50-4x)/8 = (25-2x)/4

Answer 2

A = x^2 + 3y^2
A = x^2 + 3((25/2x)/4)^2

Answer 3

looks good so far
now use differentiation

Answer 4

yeah, that's what I need help with...
I've tried it twice, got different answers both times, and was wrong both times xD
ehhh
I think I might be missing a chain rule or something
can you help?

dA/dx = 2x + 3(2)((25-2x)/4)(-1/2) ?

Answer 5

in your last line that should be 25-2x - looks like a typo

Answer 6

yes, sorry, (25-2x)/4

Answer 7

well you could expand it out and get a staight forward differentiation - it would be a bit messy though

Answer 8

lol - i just seen an error - thats not right!

Answer 9

tah dah xD what is it?

Answer 10

no - in my work - i did it by expanding first

Answer 11

lol

Answer 12

I mean, my ti 89 told me the answer to the question is 66.964...
but I'm supposed to be doing this mostly by hand... soooo

Answer 13

i cant see anything wrong so far
you've applied the chain rule to the second term and it looks correct

Answer 14

though it looks a bit messy to solve in that form

Answer 15

if you expand you get
x^2 + 3/16 ( (25 - 2x)^2
= x^2 + 3/16( 625 -100x + 4x^2)

agreed?

Answer 16

ah ha! I messed up distributing a negative before xD

2x + (-3(25-2x)/4)
2x + (-75 + 6x)/4
(8x + 6x - 75) / 4
(14x - 75) /4
then x = 75/14

Answer 17

*does happy dance*
w00t got it, thanks for helping @cwrw238 ! <3

Answer 18

thats 5.375 right - yes i get that too

Answer 19

yw its is right - i did it a different way and got the same answer

Answer 20

after differentiating i got 3.5x = 18.75

Answer 21

make sure you "justify results"

Answer 22

yeah, doing that now @Zarkon

x A
---------------
0 | 117.1875
75/14 | 66.964
root 50 | 72.105

therefore 75/14 is the minimum area

Answer 23

why root 50?

Answer 24

I made x be the side of the square
and the maximum the side of the square can be when the ...
wait... oh
when the perimeter is 50
soooo
50/4

Answer 25

yes...50/4

Answer 26

or to show its a minimum you can find the second derivative and find its sign when x = 75/14

Answer 27

yea... that was the other option... but with the way I was deriving that stuff... it'll get too ugly I think xD

Answer 28

yes
with my way i got dA/dx = (7/4)x^2 - 18.75

so d^A/dx^2 = (14/4)x = 3.5 x

so x = (75/14) gives a positive so minimum

Answer 29

mhmm... *nods blindly* I'll take your word for it xD

I don't think it would be too bad now that I look at it... but I think I prefer plugging and chugging :)

thanks for you help! :D <3