A farmer decides to enclose a rectangular garden using the side of the barn as on of the sides of the rectangle. what is the maximum area that the farmer can enclose with 48ft of fence? what should the dimensions of the garden be to give this area?

Accepted Solution

Answer:Step-by-step explanation:Let the length of the fence = x Let the width of the fence = y  Recall that the perimeter of a rectangle is calculated by 2(L+B) , but the farmer is using the side of the barn on one side of the rectangle , so the perimeter equation is  x + 2y = 48  Area = xy  If we substitute the perimeter equation so that the area is only in terms of y.  Area = (48 - 2y)y Area = 48y - 2[tex]y^{2}[/tex]  Now just find the vertex of the parabola  Area = -2 [tex]y^{2}[/tex]+ 48y A = -2 [tex]y^{2}[/tex] + 48Differentiate A with respect to y[tex]\frac{dA}{dy}[/tex] = -4y + 48equate it to zero , we have -4y + 48 = 04y = 48 y = 12Substitute y = 12 into equation 1, we havex = 48 - 2yx = 48 - 2(12)x = 48 - 24x = 24Therefore the dimensions of the garden are 24 by 12The maximum area is 288 square unit