Problema Solution
farmer ed has 9000 meters of fencing, and wants to enclose a rectangular plot that borders on a river. if farmer ed does not fence the river, what is the largest area that can be enclosed?
Answer provided by our tutors
let
l = the length of the rectangular plot, l >0
w = the width of the rectangular plot, w>0
Case 1: one of the lengths of the plot is by the river thus the fencing perimeter is
l + 2w = 9000
l = 9000 - 2w
the area of the rectangular plot is A = l*w
if we plug the value for l = 9000 - 2w in A = l*w we get
A = (9000 - 2w)*w
A = - 2w^2 + 9000w
we need to find the maximum for A
since the quotient in front of w^2 is -2 < 0 the function has maximum equal to the vertex c - (b^2/(4a)) where a = -2, b = 9000, c =0
A max = - 9000^2/(4*(-2)) = 10,125,000 m^2
click here to see the graph of the function
Case 2: one of the widths is by the river thus the fencing perimeter is
2l + w = 9000
w = 9000 - 2l
the area of the rectangular plot is A = l*w
if we plug the value for w = 9000 - 2l in A = l*w we get
A = (9000 - 2l)*l
A = - 2l^2 + 9000l
we need to find the maximum for A
since the quotient in front of l^2 is -2 < 0 the function has maximum equal to the vertex c - (b^2/(4a)) where a=-2, b=9000, c=0
A max = - 9000^2/(4*(-2)) = 10,125,000 m^2
click here to see the graph of the function
the largest area that can be enclosed is 10,125,000 m^2.