To solve this problem, let’s define the dimensions of the rectangular garden:

• Let $x$ be the length of the garden along the side with the brick wall (measured in feet).
• Let $y$ be the width of the garden, which will be enclosed by the metal fence on the remaining three sides.

Given:

1. The area of the garden is 6400 square feet. $x \times y = 6400$ So, $y = \frac{6400}{x}$.

2. The cost per foot of the brick wall is $60, and the cost per foot of the metal fence is $20.

Step 1: Set up the cost function

The cost of enclosing the garden includes:

• The cost of the brick wall on one side (length $x$): $60 \times x$
• The cost of the metal fence on the other three sides (two widths $y$ and one length $x$): $20 \times (2y + x)$

Thus, the total cost $C$ is: $C = 60x + 20(2y + x)$

Substitute $y = \frac{6400}{x}$ into the cost function: $C = 60x + 20 \left(2 \cdot \frac{6400}{x} + x\right)$ $C = 60x + 20 \left(\frac{12800}{x} + x\right)$ $C = 60x + \frac{256000}{x} + 20x$ $C = 80x + \frac{256000}{x}$

Step 2: Minimize the cost function

To find the minimum cost, we need to take the derivative of $C$ with respect to $x$, set it to zero, and solve for $x$.

1. Differentiate $C$ with respect to $x$: $\frac{dC}{dx} = 80 - \frac{256000}{x^2}$

2. Set the derivative equal to zero to find critical points: $80 - \frac{256000}{x^2} = 0$ $\frac{256000}{x^2} = 80$ $x^2 = \frac{256000}{80}$ $x^2 = 3200$ $x = \sqrt{3200} = 56.57 \, \text{ft} \, (\text{approximately})$

3. Substitute $x = 56.57$ back into $y = \frac{6400}{x}$ to find $y$: $y = \frac{6400}{56.57} \approx 113.14 \, \text{ft}$

Solution

• The brick-walled side should be approximately 56.57 feet.
• The other dimension of the garden should be approximately 113.14 feet.

Would you like further details on this process or have additional questions?

Here are five related questions to expand your understanding:

1. What would the dimensions be if the area of the garden changed to 5000 square feet?
2. How would the cost function change if both sides had a different cost per foot?
3. What are the steps to verify if these dimensions truly minimize the cost?
4. How can we use second derivative tests to confirm minimum points in optimization?
5. Can this problem be generalized to different shapes, like circular gardens?

Tip: In optimization problems involving cost or area, setting up a function in terms of one variable simplifies the process.