Optimization? I got stuck on this one?

A landscape architect plans to enclose a 2100 square foot rectangular region in a botanical garden. She will use shrubs costing $25 per foot along three sides and fencing costing $10 per foot along the north side.


(a) If x is the length of fencing used, write a formula for the total cost in terms of x only.


(b) Find the length of fence that will minimize the total cost.


(c) Find the minimum total cost. Give your answer to the nearest penny.

Optimization? I got stuck on this one?
The area is 2100 sq. ft. Let x be the length of two of the opposite sides and y be the length of the other two opposite sides. Then x*y = 2100,





Or y = 2100 / x





The total length of the three sides in question is





x + x + 2100 / x = 2x + (2100 / x)





Thus the cost of fencing those three sides is


25 * [2x + (2100 / x) ]





The remaining side has length 2100 / x . This piece costs 10 dollars per ft., so the total cost of that side is 10 * 2100 / x


= 21000 / x





Thus the total cost to build the fence in terms of x only is


f(x) = 25 * [2x + (2100 / x) ] + 21000 / x





or f(x) = 50x + (52500 / x) + (21000 / x)


= 50x + (73500 / x)





To optimize it, take the derivative and set it equal to 0:





f '(x) = 50 - (73500 / (x^2)) = 0





And solve for x:





x = 7*sqrt(30) = approximately 38.34 feet for the two sides of shrubs





since x = 1300 / y, plug in x = 7*sqrt(30) to get





y = (130/21)*sqrt(30) = approximately 33.91 feet on the the front row and bushes and the back fence.





The minimum cost is when x = 7*sqrt(30), plug that into f(x) and evaluate it for the answer.