Maths calculus i think?

Question

A farmer plans to use a river as one boundary of a rectangular paddock. If the farmer has 960m of fencing to be used to fence the OTHER 3 SIDES, what dimensions should the paddock be to ensure maximum area?

Please show working for the answer as well.
its for revision and i have final answer but been trying to solve it 4 bout 15mins with no luck.
its probably really simple but i just cant get my head around it.

Answer 1

let the length of the paddock be 'a' & breadth be 'b'.
let the side common with river be equal to 'a'..
so
960=a + 2*b...........(1)
area,A=a*b
using eqn (1)
A=(960-2*b)*b
differentiating w.r.t 'b'
d(A)/db=960-2*b+(-2)*b=0 (for maxima)

4b=960
b=240

putting value of 'b' in (1)
a=960-2*240
a=480

so dimensions are 240 & 480

Answer 2

Well, I hope you know that a circle takes the maximum circumference and gives the minimum area (that's why in 3-d raindrops etc are spherical: max surface area). The figure that'll give you the maximum area is a square (maximum symmetry applied to a quadrilateral, actually). If you increase the sides, it tends to become more and more semi-circular in shape, hence reduces the area. Find out why a square will give more area than a triangle then!

Answer 3

Let the side perpendicular to the river bank be x.
The long side is 960 - 2x
The area is:
x(960 - 2x)
Take the derivative and set it equal to zero:
-4x + 960 = 0
x = 240
960 - 2x = 480

Answer 4

Let fencing have sides x ,x and 960 - 2x
A(x) = x.(960 - 2x)
A(x) = 960 x - 2x²
A `(x) = 960 - 4x = 0 for maximum A
x = 240
Sides are 240m, 240m and 480m

Answer 5

I had a question similar to that on my level 3 exam, we all lost marks for assuming it was a square, I still dont know how else to do it so cant really help