How to find the dimensions that will require the least amount of fencing?

A rectangular garden 450 square feet in area is to be fenced off against hyenas. Find the dimensions that will require the least amount of fencing if one side of the garden is already protected by a barn.


Can anyone help me solve it! Thank you?|||You're trying to minimize L, the length of fencing. Since one side of the fenced area is covered by the barn, you've got to make two sides perpendicular to the barn plus one side parallel to the barn. Let the length of the perpendicular side be x. Let the length of the parallel side be y. Thus





x*y = 450


2*x + y = L





Substitute y = 450/x into the second equation, and you get





2x + 450/x = L





To find the length x that minimizes L, take the first derivative of L and set it equal to zero to find the extrema.





2 - 450/x^2 = 0 ==%26gt; 2 = 450/x^2 ==%26gt; x^2 = 450/2 = 225 ==%26gt; x = 15. (or -15, but that is an extraneous solution).





If x = 15, y = 450/x = 30. Thus, L = 2x + y = 2*15 + 30 = 60 feet of fencing.





Edit: technically, to make sure this is a minimum, you should evaluate the second derivative of L, which in this case is 900/x^3, which is greater than 0 for all x %26gt; 0. If your second derivative is positive at the extreme point (which it is at x = 15), then you indeed have a minimum.