A farmer plans to use 21 m of fencing to enclose a rectangular pen having an area of 55 m². Only three sides of the pen need fencing because of an existing wall. Find the dimensions of the pen?

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Gaurang Vaishnav Profile
Let the length of the sides of pen at right angle to the wall be equal to x meters.  Let the length of the sides of pen  parallel to the wall be equal to y meters.  Then the length of the fence needed is 2x +y; this is equal to 21m.  Therefore, 2x+y = 21  Area of the pen is x times y  = xy; this is 55 square meter.  Therefore, xy = 55  2x+y = 21, therefore, y = 21-2x  Substituting value of y in the equation xy = 55,we have x(21-2x) = 55therefore, 21x-2x2  = 55  therefore, 0=55-21x+2x2therefore, 2x2 -21x+55 =0This can also be put as: (2x-11) x (x-5) = 0 Since the product of two values , i.e., (2x-11) and (x-5) is zero, either both of them or alt least one of them is equal to zero.if 2x - 11 = 0 then, x = 11/2if x-5 = 0 then, x = 5Since 2x+y = 21, if x =11/2, then y = 21-2x11/2 = 10 If x = 5, then y = 21-2x5 = 11 Therefore, there are two possible answers. Pen is 5 1/2 x 10 or 5 x 11.

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