To solve this problem, we will first define the variables and write the constraints based on the given information.

Step 1: Defining Variables

• $x$: The length of the vertical side of the garden (along the wall).
• $y$: The length of the horizontal side of the garden (perpendicular to the wall).

The total amount of wire is 120 feet. Since the vertical side needs 4 strands of wire and the horizontal side needs 5 strands, we can write an equation for the total wire used:

$4x + 5y = 120$

Step 2: Objective Function

We want to maximize the area of the garden, which is given by:

$A = x \cdot y$

Step 3: Solve for One Variable in Terms of the Other

We can express $y$ in terms of $x$ using the constraint equation:

$5y = 120 - 4x$ $y = \frac{120 - 4x}{5}$

Step 4: Substituting into the Area Formula

Substitute the expression for $y$ into the area formula:

$A = x \cdot \left( \frac{120 - 4x}{5} \right)$ $A = \frac{x(120 - 4x)}{5}$ $A = \frac{120x - 4x^2}{5}$

Step 5: Maximizing the Area

To maximize the area, we will take the derivative of $A$ with respect to $x$ and set it equal to zero.

$\frac{dA}{dx} = \frac{d}{dx} \left( \frac{120x - 4x^2}{5} \right)$ $\frac{dA}{dx} = \frac{120 - 8x}{5}$

Set the derivative equal to zero to find the critical points:

$\frac{120 - 8x}{5} = 0$ $120 - 8x = 0$ $8x = 120$ $x = 15$

Step 6: Find $y$

Now substitute $x = 15$ back into the equation for $y$:

$y = \frac{120 - 4(15)}{5}$ $y = \frac{120 - 60}{5}$ $y = \frac{60}{5}$ $y = 12$

Final Answer

The values of $x$ and $y$ that maximize the area are:

$x = 15 \quad \text{and} \quad y = 12$

These dimensions will enclose the maximum area.

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Would you like further details on how to approach similar problems or any clarifications on this one? Here are some related questions to explore:

1. How can you generalize this type of optimization problem for other shapes (e.g., circular or triangular fences)?
2. What is the significance of the second derivative in confirming the maximum area?
3. How would the solution change if the amount of wire were different?
4. How does the number of strands for each side affect the final dimensions?
5. How can this approach be used in real-life scenarios, like fencing a garden or property?

Tip: When solving optimization problems with constraints, always start by expressing the constraint in terms of one variable and substitute it into the objective function before differentiating.