Question

A farmer has 1,200 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. Write the function that will produce the largest area if x is the short side of the rectangle.
f(x) = 1200x − x2
My answer choice. Is this correct? ->f(x) = −2x2 + 1200x
f(x) = x2 − 1200
f(x) = 2x2 − 1200

Answer 1

You are correct.  It would be
f(x) = x(1200-2x) which equates to your selection.

Answer 2

Yay, derivitives
I'mma ignore that x is the shorter side because I don't know which one has to be shorter yet


we need to find the max area
but with 3 sides

area=LW
let's say the sides are z and y
zy=area

and the relatiionship between them is
hmm,
z+2y=1200
because one side has no fencing
so
z+2y=1200
solve for z
z=1200-2y
sub for z in other

(1200-2y)(y)=area
expand
1200y-2y²=area
take derivitive
1200-4y=dy/dx area
max is where dy/dx goes from positive to negative
solve for where dy/dx=0
1200-4y=0
1200=4y
300=y

at y<300, dy/dx<0
at y>300, dy/dx>0
so at y=300, that is the max

then
z=1200-2y
z=1200-2(300)
z=1200-600
z=600

so then
z=600
y=300
300<600

so the shorter side would be y

so then we see our choices and noticed that
erm
I think it is f(x)=1200x-2x²
takind the derivitive yeilds none of the others


so ya, you are right