A gardener has 120 feet of fencing to fence in a rectangular flower garden. a) Find a function that models the area of the garden she can fence. b) Can she fence a garden with an area of 850 square feet? c) Find the dimensions of the largest area she can fence.

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A gardener has 120 feet of fencing to fence in a rectangular flower garden. a) Find a function that models the area of the garden she can fence. b) Can she fence a garden with an area of 850 square feet? c) Find the dimensions of the largest area she can fence.

Mathematics
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(a) P=2l+2w Perimeter=P=120 length=l width=w 120=2l+2w
oops Area=A and since 120=2l+2w 60=l+w so w=60-l so A=lw=l(60-l)=60l-l^2
there now a is complete

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Other answers:

Okay, that makes sense.
part b now) If A=850 then 850=60l-l^2 l^2-60l+850=0 now we need to find l here
Oh, so it's a quadratic equation...
right i don't think we can factor it but you use you use the quadratic formula
what did you get?
l=37.07 or l=22.93
ok so if l=37.07 then w=60-37.07 which is possible since the w is positive and if l=22.93, then is also possible since w is positive if we have this length
So she can fence in a garden of 850 sq. feet?
are you sure you got l=37.07 and 22.93
Yeah...
a=1 b=-60 c=850 ?
\[\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
\[l=(60\pm \sqrt{(-60)^2-4(1)(850)}/2(1)\]
\[\frac{60 \pm \sqrt{60^2-4(1)(850)}}{2}\]
Yeah.
i got the lengths to be appoximately 52.93 and 67.07
It would be \[60\pm14.14/2\]right?
she can fence the garden as long as the length is less that 60 we have 52.93 so it is possible to fence an area of 850 square feet
let me try again
oops you are right lol
ok so remember from before w=60-l?
we want w and l to be positive because negative lengths don't exist
we want w to be positive so we want 60-l>0 so 60>l (l<60)
so do you think it is possible since both of the lengths we got are less than 60?
and greater that 0?
than*
does that make sense? 0
Yeah, it does. Sorry I jsut ate dinner :P
ok find the vertex of A and you will find the maximum area put the A (which is a parabola) in vertex is form
A=60l-l^2 A=-l^2+60l A=-(l^2-60l) A=-(l^2-60l+(60/2)^2)+(60/2)^2 A=-(l^2-60l+30^2)+30^2 A=-(l-30)^2+900 the vertex is (30,900) so max length is 30 and max area is 900
Ohhhh okay :D

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