Problema Solution
A farmer wants to fence in land next to a river. Because the river is there, only 3 sides need to be fenced. The farmer has 200 meters of fence to use. What is the maximum area we can have inside his fence
Answer provided by our tutors
lets assume that the farmer has a rectangular piece of land with
l = the length of the land, l>0
w = the width of the land, w>0
if one of the lengths is by the river the farmer needs to fence a perimeter of l + 2w
l + 2w = 200 => l = 200 - 2w
the area inside the field is A = l*w and if we plug l = 200 - 2w we get the equation
A = (200 - 2w)*w
A = - 2w^2 + 200w
A has maximum since -2 <0 (the quotient in front on w^2) is in the vertex and it is equal to 200^2/4*2 = 5000 m^2. (c - (b^2/4a))
we get the same result if one of the width is by the river.
the maximum area we can have inside the fence is 5000 m^2.