Question

A farmer has 1900 feet of fencing available to enclose a rectangular area bordering a river. of no fencing is required along the​ river, find the dimensions of the fence that will maximize the area. what is the maximum​ area

Answer 1

ya, calc

so lets say we have x and y are the length so f the sides
x is the side that is across from the river
y is the 2 sides

so the perimiter is x+2y=1900

the area is xy

so

x+2y=1900
solve for x
x=1900-2y
subsitute that for x in other equaiton

(1900-2y)(y)=area
1900y-2y²=area
take derivive or find the vertex
I like deriviives so whatever

1900-4y=dy/dx area
where does it equal 0?
1900-4y=0
1900=4y
divide both sides by 4
475=y

is it a max? if it is then the slope changes from positive to negative there
dy/dx area>0 when y<0
dy/dx area<0 when y>0

so it is a max
y=475
sub back
x=1900-2y
x=1900-2(475)
x=1900-950
x=950

the dimentions are 950ft by 950ft
the max area=950 times 950=902500 square feet