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.

East and west side = 5 foot

South and north side = 10 foot

Area = 50 foot^2

Step-by-step explanation:

We know that:

East and west fencing cost = $4 per foot

South and north fencing cost $2 per foot

So, we can consider:

x = south and north fencing

y = east and west fencing

Then we create an equation representing the case:

Total cost = (south+north) * $2 + (east+west) * $4

80 = (x+x) * 2 + (y+y) * 4

80 = (2x) * 2 + (2y) * 4

80 = 4x + 8y

80 = 4 * (x+2y)

80/4 = x+2y

20 = x+2y

x = 20-2y

We can calculate the area as width * length:

Area = length * width

Area = x * y

Area = (20 - 2y) * y

area = 20y - 2y^2

Next 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 - 4y

20 - 4y = 0

20 = 4y

y = 20/4

y = 5

If y = 5, then:

x = 20 - 2y

x = 20 - 2*5

x = 20 - 10

x = 10