Claire wants to fence in a rectangular plot of land along a

Claire wants to fence in a rectangular plot of land along a wall to make a garden. She has p feet of fencing, which will extend along three sides of the garden, with the wall forming the fourth side, as shown in the figure above. Which of the following expressions p is equal to the maximum possible area, in square feet, of Claire\'s garden?

Solution

Let the length of the rectangular garden be x feet and let its width be y feet. Since p feet of fencing extends along three sides ( length +2*width), we have x+2y = p or, y = (p-x)/2. Further the area of the rectangular garden is length *width = x*y = x*(p-x)/2 = -x²/2 + px/2. We know that A will be maximum when dA/dx = 0 and d²A/dx² is negative. Here, dA/dx = -x +p/2 which is zero when x = p/2. Also, d²A/dx² = -1 is negative regardless of the value of x. Hence the area of the rectangular garden will be maximum when x = p/2. Then theis maximum area is x*(p-x)/2 = p/2(p-p/2)/2 = (p/2)*p/2 *1/2 = p²/8. Option D is the correct answer.