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

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 area0

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