Problema Solution
A farmer plans to use 21 meters of fencing to enclose a rectangular pen having area 55 meters^2. Only three sides of the pen need fencing because part of the existing wall will form the fourth side. Find the dimensions of the pen.
Answer provided by our tutors
let
l = the length of the rectangular pen, l>0
w = the width of the rectangular pen, w>0
l > w
the area is 55 m^2
l*w = 55
farmer plans to use 21 meters of fencing to enclose a rectangular pen and only 3 sides need fencing thus
we have 2 cases:
1) one length doesnt need fencing then the perimeter is 2w + l = 21
2w + l = 21 => l = 21 - 2w and plug in l*w = 55
(21 - 2w)w = 55
by solving the quadratic equation we find
w1 = 5 m => l1 = 11 m
w2 = 5.5 m => l2 = 10 m
the dimensions of the pen are 5 m and 11 m or 5.5 m and 11.5 m.
click here to see the solution of the quadratic equation
2) one width doesnt need fencing then the perimeter is w + 2l = 21
w + 2l = 21 => w = 21 - 2l and plug in l*w = 55
(21 - 2l)l = 55
by solving the quadratic equation we find
l1 = 5 m => w1 = 11 m
l2 = 5.5 m => w2 = 10 m
since the length > width the rectangular pen must have one of its lengths by the wall.