4. A rectangular garden is to be surrounded by 4 sides of a brick wall costing $$ \$ 20 / \mathrm{ft}, \$ 10 / \mathrm{ft}, \$ 9 / \mathrm{ft} $$, and $$ \$ 8 / \mathrm{ft} $$ for the sides. The $$ \$ 20 $$ side must be next to the $$ \$ 10 $$ side. Find the maximum area when you only have a budget of $$ \$ 5200 $$. [8] (2dp)

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To solve this problem, we need to maximize the area of a rectangular garden surrounded by a brick wall with different costs for each side. Let's denote the lengths of the sides of the rectangle as follows:

- Let $$ x $$ be the length of the side that costs $$ \$ 20/\mathrm{ft} $$.
- Let $$ y $$ be the length of the side that costs $$ \$ 10/\mathrm{ft} $$.
- The opposite side of $$ x $$ will also cost $$ \$ 20/\mathrm{ft} $$.
- The opposite side of $$ y $$ will also cost $$ \$ 10/\mathrm{ft} $$.
- The remaining two sides will cost $$ \$ 9/\mathrm{ft} $$ and $$ \$ 8/\mathrm{ft} $$.

### Step 1: Set up the cost equation

The total cost of the walls can be expressed as:
$$
\text{Total Cost} = 2(20x) + 2(10y) + 9y + 8x
$$
This simplifies to:
$$
\text{Total Cost} = 40x + 20y + 9y + 8x = 48x + 29y
$$

### Step 2: Set up the budget constraint

Given that the budget is $$ \$ 5200 $$, we have:
$$
48x + 29y \leq 5200
$$

### Step 3: Set up the area function

The area $$ A $$ of the rectangle is given by:
$$
A = x \cdot y
$$

### Step 4: Solve for $$ y $$ in terms of $$ x $$

From the budget constraint, we can express $$ y $$ in terms of $$ x $$:
$$
29y \leq 5200 - 48x
$$
$$
y \leq \frac{5200 - 48x}{29}
$$

### Step 5: Substitute $$ y $$ into the area function

Substituting $$ y $$ into the area function gives:
$$
A = x \cdot \left(\frac{5200 - 48x}{29}\right)
$$
$$
A = \frac{5200x - 48x^2}{29}
$$

### Step 6: Maximize the area function

To find the maximum area, we can take the derivative of $$ A $$ with respect to $$ x $$ and set it to zero:
$$
\frac{dA}{dx} = \frac{5200 - 96x}{29}
$$
Setting the derivative equal to zero:
$$
5200 - 96x = 0
$$
$$
96x = 5200
$$
$$
x = \frac{5200}{96} \approx 54.17
$$

### Step 7: Find $$ y $$

Now, substitute $$ x $$ back into the equation for $$ y $$:
$$
y = \frac{5200 - 48(54.17)}{29}
$$
Calculating $$ y $$:
$$
y = \frac{5200 - 2600.16}{29} \approx \frac{2599.84}{29} \approx 89.65
$$

### Step 8: Calculate the maximum area

Now we can calculate the maximum area:
$$
A = x \cdot y \approx 54.17 \cdot 89.65 \approx 4855.57
$$

### Conclusion

The maximum area of the rectangular garden, given the budget constraints and costs of the walls, is approximately $$ 4855.57 \, \text{ft}^2 $$.

Thus, the final answer is:
$$
\text{Maximum Area} \approx 4855.57 \, \text{ft}^2
$$
The maximum area of the garden is approximately 4855.57 square feet.

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