To minimize the cost, let's set up the problem and find the dimensions.

Given:

• Area of the garden = $x \cdot y = 22$ square feet.
• One side (length $x$) is enclosed by a brick wall costing $40/ft, and the other three sides are enclosed by a metal fence costing $10/ft.

1. Cost Function: The total cost $C$ is calculated by adding the cost of the brick wall and the metal fence: $C = 40x + 10(2y + x)$ Simplifying this, we get: $C = 50x + 20y$

2. Express $y$ in terms of $x$: Since the area is fixed at 22 square feet, we have: $y = \frac{22}{x}$

3. Substitute $y$ in the Cost Function: $C = 50x + 20\left(\frac{22}{x}\right)$ Simplifying further, we get: $C = 50x + \frac{440}{x}$

4. Minimize the Cost Function: To find the minimum cost, take the derivative of $C$ with respect to $x$ and set it to zero. $\frac{dC}{dx} = 50 - \frac{440}{x^2}$ Setting this equal to zero: $50 = \frac{440}{x^2}$ Multiply both sides by $x^2$: $50x^2 = 440$ $x^2 = \frac{440}{50} = 8.8$ $x = \sqrt{8.8} \approx 2.97$

5. Find $y$: $y = \frac{22}{x} = \frac{22}{2.97} \approx 7.41$

So, the dimensions of the garden that minimize the cost are approximately:

• Length of side with bricks $x \approx 2.97$ feet
• Length of adjacent side $y \approx 7.41$ feet

Would you like further details on any step or have any questions?

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1. What happens to the cost if the area of the garden increases?
2. How would the solution change if the brick wall cost was different?
3. What if the fence cost was higher or lower?
4. Can we use different methods to find these dimensions?
5. How does changing the shape of the garden (keeping area constant) affect cost minimization?

Tip: When optimizing costs, setting up the function correctly is crucial. Make sure each element's cost is accurately represented in the equation.