Question

A rectangular field is to be enclosed by 800 m of fencing. What dimensions will give the maximum area?

Answer 1

let A be one side B the other of the rectangular field.
than we have that 2(a+b)=800
and we want to find the max for a*b

Answer 2

have you learnt differentiation?

Answer 3

Er, I don't think so. This is in the Quadratic functions unit

Answer 4

oh ok, so from the 2(a+b)=800
a+b=400
a=400-b
now put this back to the area
a*b=b(400-b)

Answer 5

is this clear until this, sorry that I am a bit slow, doing something else too :-)

Answer 6

why does a*b =b(400-b)

Answer 7

because a=400-b
so we can swap the a for 400-b

Answer 8

Okay. So what do I do next?

Answer 9

? Where did the post go?

Answer 10

now we want to find the max for b(400-b)=400b-b^2

Answer 11

nevermind!

Answer 12

I deleted it because it was wrong :-)

Answer 13

do you know how to find the maximum of a function like this?

Answer 14

um. no.
and I thought a*b = 400b-b^2

Answer 15

yeah thats right and we need to find for which value(s) of b will it have a max

Answer 16

okay. I'm confused. Go on though :p

Answer 17

Wait, isn't b(400-b)=400b-b the same as 400b-b^2=400b-b^2 ??/

Answer 18

ok you have to rewrite this expression to make a perfect square
400b-b^2= - (b^2-400b)= -((b-200)^2 -40000)

Answer 19

first step I took out the - so b^2 is positive

Answer 20

2nd step is to make a perfect square
b^2-400b
think about expanding the brackets for this (x+y)^2=x^2+2xy+y^2
so it will be (b+200)^2 as if you expand the bracket you get
b^2+400b+40000
now that is 20000 more than our expression so we need to subtract that
so b^2-400b=(b+200)^2 -40000

Answer 21

there is a typo, not 20000 but 40000

Answer 22

and now we are nearly done

Answer 23

because we want to find the maximum for this:
-((b-200)^2 -40000)=

=40000- (b-200)^2

Answer 24

do you see the solution?

Answer 25

OKAY, I had to write that all out for it to make sense, but in the end it did, :$. Thanks!

Answer 26

That is great!