A landscape architect plans to enclose a 2,500 square feet rectangular region In a botanical garden. She will use shrubs costing $18 per foot along three sides and fencing costing $32 per foot along the fourth side. Find the dimensions of the botanical garden that will minimize the total cost. Follow the steps: (a) Let the width (the fencing side) to be y and the length to be x . Then the quantity to be minimized is (expressed as a function of both x and y] C: 32y + 54.x x 2500 X (b) The condition that x and 3; must satisfy ls y: (c) Using the condition to replace y by x In C, C can then be expressed as function of x: C(x)= (d) The domain of C |s( 0 J , infly J 3. (Um ' 'infty" For an.) (e) The only critical number of C in the domain is xE O x . (Keep 1 decimal place (roundedn. We use the Second-Derivative nest to classify the critical number as a relative maximum or minimum, or neither: At the critical number xa 0 x , the second derivative C"( 0 x )is positive a J . Therefore at xa 0 X , the function has a relative minimum ( (f) Fina ly, plug x: 0 x Into the condition of x and y we obtain )1: 0 x . Therefore the length and width of of the botanical garden that will minimize the total cost for materials are XE 0 x feet and ya I} x feet, with the fencing side equals 42.4 J feet. Z'symbolic lnrmatting help