To minimize the cost of the fencing for the vegetable patch, we need to consider the cost of fencing for both the north-south and east-west sides and set up equations accordingly.

Let's define:

• $x$ = length of the north and south sides (in feet)
• $y$ = length of the east and west sides (in feet)

From the problem, we know:

1. The area of the patch is 162 square feet, so: $x \cdot y = 162$
2. The cost of the fencing per foot is:
   • $2 per foot for the north and south sides (total length: $2x$)
   • $4 per foot for the east and west sides (total length: $2y$)

The total cost $C$ of the fence is: $C = 2(2x) + 4(2y) = 4x + 8y$

We want to minimize the cost $C$ subject to the constraint $xy = 162$. To do this, we can express $y$ in terms of $x$ using the area constraint: $y = \frac{162}{x}$

Now, substitute this into the cost function: $C(x) = 4x + 8\left(\frac{162}{x}\right) = 4x + \frac{1296}{x}$

To find the value of $x$ that minimizes the cost, we take the derivative of $C(x)$ with respect to $x$ and set it equal to 0: $\frac{dC}{dx} = 4 - \frac{1296}{x^2} = 0$ $4 = \frac{1296}{x^2}$ $x^2 = \frac{1296}{4} = 324$ $x = \sqrt{324} = 18$

Now that we know $x = 18$, we can find $y$ using the area equation: $y = \frac{162}{18} = 9$

Thus, the dimensions of the vegetable patch that minimize the cost are:

• North and south sides (x): 18 feet
• East and west sides (y): 9 feet

Would you like further details or have any questions?

Related Questions:

1. How do you determine whether a critical point is a minimum or maximum?
2. What is the total cost of the fence with these dimensions?
3. How does the cost function behave as $x$ approaches 0 or infinity?
4. What if the costs per foot for fencing were equal on all sides—how would that affect the solution?
5. Can this type of optimization problem be solved using another method, such as Lagrange multipliers?

Tip:

Whenever optimizing cost or area, always double-check that the critical points satisfy the original constraints of the problem.