To find the dimensions of a rectangular garden that can be enclosed with 20 meters of fencing and has the maximum area, we can use the following steps:

Step 1: Express the Perimeter

Let the length of the garden be $L$ meters and the width be $W$ meters. The perimeter of the rectangle is given by: $2L + 2W = 20 \text{ meters}$ This simplifies to: $L + W = 10 \text{ meters}$

Step 2: Express the Area as a Function of One Variable

The area $A$ of the rectangle is given by: $A = L \times W$ Using the relationship $L + W = 10$, we can express $W$ in terms of $L$: $W = 10 - L$ Substitute this into the area formula: $A = L \times (10 - L)$ This simplifies to: $A = 10L - L^2$ This is a quadratic equation in $L$.

Step 3: Maximize the Area

To find the maximum area, we can take the derivative of the area function with respect to $L$ and set it equal to zero: $\frac{dA}{dL} = 10 - 2L$ Set the derivative equal to zero to find the critical point: $10 - 2L = 0$ Solve for $L$: $L = 5 \text{ meters}$ Using $L = 5$ meters in $W = 10 - L$, we get: $W = 5 \text{ meters}$

Step 4: Verify Maximum Area

The quadratic function $A = 10L - L^2$ opens downward (since the coefficient of $L^2$ is negative), so the critical point $L = 5$ meters gives the maximum area.

Step 5: Conclusion

The dimensions of the garden that maximize the area are $5 \times 5$ meters, making the garden a square.

The maximum area that can be enclosed is: $A = 5 \times 5 = 25 \text{ square meters}$

Would you like more details or have any questions?

Here are 5 related questions:

1. What would be the maximum area if you had 30 meters of fencing instead?
2. How does the shape of the rectangle affect the area for a fixed perimeter?
3. What is the minimum area that can be enclosed with 20 meters of fencing?
4. How can you generalize this approach for different perimeters?
5. What would happen if the garden were required to be a specific ratio, like 2:1, in length to width?

Tip: The maximum area for a given perimeter of a rectangle is achieved when the rectangle is a square.