geometry
posted by Anonymous .
You have 32 feet of fencing and want to determine the maximum area that can be enclosed with the fencing to make a flower garden. You do not want to deal with fractions or decimals, so you are limiting the dimensions to whole numbers. Also, the area is to be rectangular (or square) in shape.

not much of a problem here ...
the largest rectangle is obtained when that rectangle is a square, so
4 equal sides equal 32 feet,
each side is 8 feet
area is 64 square ft 
Considering all rectangles with a given perimeter, which one encloses the largest area?
The traditional calculus approach would be as follows.
Letting P equal the given perimeter and "x" the short side of the rectangle, we can write for the area A = x(P  2x)/2 = Px/2  x^2.
Taking the first derivitive and setting equal to zero, dA/dx = P/2  2x = 0, x becomes P/4.
With x = P/4, all four sides are equal making the rectangle a square.
.....The short side is P/4.
.....The long side is (P  2(P/4))/2 = P/4.
Therefore, it can be unequivicaly stated that of all possible rectangles with a given perimeter, the square encloses the maximum area.
Respond to this Question
Similar Questions

algebra 1
Charles wants to build a vegetable garden such that three sides of the garden are fenced and the fourth side of the garden will be the existing back fence. He has 30 feet of fencing available. Find the dimensions of the garden that … 
Algebra and Trig
Aldo has feet of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing. What is the maximum area that the garden can have? 
Math
A rectangular garden is to be fenced off and divided into 3 equal regions by fencing parallel to one side of the garden. 200 feet of fencing is used. find the dimensions of the garden that maximizes the total area enclosed. what is … 
Math OPTIMIZATION
A home gardener plans to enclose two rectangular gardens with fencing. The dimensions of the garden: x by 12x, y by 12xy a. Find the values of x and y that maximize the total area enclosed. b. What is the maximum total area enclosed? 
precalc
area of a rectangular region: a farmer wishes to create two rectangular regions bordering a river, by three fences perpendicular to the river and one connecting them. suppose that x represents the length of each of the three parallel … 
Polynomials and quadratic application
I want to make a rectangular garden by fencing in a area on the back of my house. Three sides of the rectangle will be fencing; the fourth side will be the side of the house. I have 60 feet of fence. What dimensions should I make the … 
Maath:)
Sam is going to plant a rectangular flower garden next to his house. There will be a fence on three sides of the garden( fencing is not necessary along the houe). The area of the garden is to be 1,000sq. ft., and he has 90 feet of … 
Math
A rancher has 220 feet of fencing to enclose a rectangular corral. Find the dimensions of the rectangle that maximize the enclosed area, using all 220 feet of fencing. Then find the maximum area. 
Math
A rectangular garden is surrounded by a roll of chicken wire fencing to keep out the rabbits. The dimensions of the garden are 2.6 m by 5.1 m. The area enclosed by the fence is ______ square metres. The length of the fence is ____ … 
geometry
A garden is in the shape of a rectangle 26 feet long and 35 feet wide. If the fencing costs $5 a foot, what will it cost to place fencing around the perimeter of the garden?