You want to fence in a rectangular vegetable patch. The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. You have a budget of $80 for the project. Use Lagrange multipliers to find the dimensions of the vegetable patch with the largest area you can enclose.

Answer:East and west side= 5 footSouth and north side = 10 footArea = 50 foot^2Step-by-step explanation:We know that:East and west fencing cost= $4 per footSouth and north fencing cost $2 per footSo, we can consider:x= south and north fencingy= east and west fencingThen we create an equation representing the case:Total cost = (south+north)*$2 + (east+west)*$480 =(x+x)*2 + (y+y)*480 =(2x)*2 + (2y)*480 =4x + 8y80 =4*(x+2y)80/4 =x+2y20 = x+2yx= 20-2yWe can calculate the area as width * length:Area = length * widthArea= x * yArea= (20 - 2y) * yarea= 20y - 2y^2Next step is to find the "y" value for the maximum area so you can derivate and equal to 0 to find maximums:d(20y - 2y^2)/dy = 20 - 2*2y = 20 - 4y20 - 4y = 020 = 4yy = 20/4y = 5If y = 5, then:x = 20 - 2yx = 20 - 2*5x = 20 - 10x = 10