What is the maximum area Reena can enclose if she has 30 m of fencing?

Reena wants to fence part of her front yard for a flower bed.

What is the maximum area Reena can enclose if she has 30 m of fencing?
Let the length of plot be l and width be w ...

then

Length of fencing is equal to perimeter of rectangular flower bed



so this means 2l + 2w = 30

or 2l = 30 - 2w

l = 15 - w





Area of feild = length x width = l *w

= (15 -w ) *w

= 15 w - w^2

= -w^2 + 15 w



use property of QUADRATIC EQUATION ( ax^2 + bx + c = 0 )

the maximum value is given by x = -b/2a



so here we get w = -15/2(-1) = 7.5

then length become l = 15 - 7.5 = 7.5

so length = width ....





so this means the plot is in square shape with maximum

area = 7.5* 7.5 square unit = 56.25 m square.
Reply:If the fence isn't flexible then the greatest area would be a square.



30/4=7.5



7.5*7.5=56.25m2



However if she can bend the fence then the greatest area will be a circle.



If the circumference is 30m then:



30/Pi=9.55



This is the diameter.



Diameter/2=4.77



4.77*4.77*Pi=71.61972439m2 or 71.62m2



This is the largest possible area that can be enclosed.
Reply:Assuming that the area is rectangular...



Set up the equation for perimeter where x is length and y is width. 2x+2y=30. Solve for x to get that x=15-y. Plug this into the formula for area: x*y = (15-y)*y = 15y - y^2. To obtain max area, first derive this formula: d/dy(15y - y^2) = 15 - 2y. Set this derivative equal to zero: 15 - 2y = 0, y = 15/2. Plug this y value into the perimeter equation to obtain the x value: 2x+2(15/2)=30, x=15/2. Finally to answer the question, solve for area: x*y= (15/2)*(15/2) = 225/4 = 56.25 square meters.
Reply:The maximum area that can be enclosed will be circular in shape.



The circumference of the circle is 30 m.



So the radius will be 30/2Π = 4.8 m (approx.)



Thus the area enclosed will be Π*(square of radius)= 72.4 m.



This is the maximum area that can be enclosed by a fencing of 30 m.



PS. If u have a doubt that why the maximum area will be a circle, you can check by taking the are as rectangle, square etc. and compare the results which u get.
Reply:56.25 for a square bed (length * Width)



71.6183m for a circular bed (Circumference = 30 = 2 * 3.14 * r^2 to find the radius, then Area = 3.14 * r^2)
Reply:circle always will have the largest area



so



(30/3.14/2)*(30/3.14/2)*3.14 or 71.65 square metres