I have a word problem that reads as so
A farmer wants to fence a rectangular part of his land (30,000 square feet). The fenced area is to have one border shared with a neighbor which he wishes to be fancy so he will spend 25 dollars per linear foot on that side where as he will use 5 dollars per linear foot of fencing on the rest. Find the dimensions that will minimize the cost
I set up my equations (x shares the border with the fancy fence)
$ A = xy $
$ C = 25x + 5(x + 2y) = 30x + 10y $
Then I found my x-value
$ x = \frac{30,000}{y} $
Plugged it in
$ C = 30(\frac{30,000}{y}) + 10y = \frac{900,000}{y} + 10y $
Derivative of C
$ C' = \frac{900,000}{y^2} + 10 $
$ -900,000 = -10y^2 $
$ y^2 = 90,000 $
$ y = 300 $
Finally, plugged it into original equation
$ 30,000 = 300x = x = 100 $
I just wanted to check if I've done this right. If not, can you explain where I've gone wrong?