A woman has 100 ft of fencing to enclose a rectangular garde

A woman has 100 ft of fencing to enclose a rectangular garden. Find the dimensions of the garden that would enclose the largest area. What is the largest area? (for your answer enter only the numbers without units)

Dimensions: length feet ;

width feet;

the largest area is;

Solution

Given

A woman has 100 ft of fencing to enclose a rectangular garden.

this means the perimeter of the garden is 100 ft.

perimeter of a rectangullar garden = 2(L+B)             //(B is bredth and l is length)

therefore

100 = 2(L+B)

we get

L+B = 50                    (1)

B = 50 - L                    (2)

now the area of a rectangle is L*B

A=L*B

A = L*(50-L)

A = 50L - L^2

A = - L^2  + 50 L

A= -1 (L^2 - 50 L)

now we have to make \"(L^2 - 50 L)\" as a perfect square.

so we divide the coefficient of L by 2 we get \"25\'\'and square it we get 625

so we add and subtract 625

A= -1 (L^2 - 50 L + 625 - 625)

now \"L^2 - 50 L + 625\" is a perfect square of (L-25) so we move out -625 outside the bracket.we get

A= -1 (L^2 - 50 L + 625 ) + 625                (since -1 * -625 = 625) so 625 comes out from the bracket

A = -1(L-25)^2 + 625

for this area to be maximum (L-25)^2 should be zero for this L should be equal to \"25\"          //(if we choose anything except \"25\" we will get a number (L-25) that will be either positive or negative, after that we square that number becomes positve . since -1 is multiplied with this so we get a negative number so the area wiill decrease)

so the length is 25

from (1)

L+B =50

we get B = 25

Area = 625