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The problem involves optimizing the area of a right-angled triangular garden along a stream, where fencing costs differ for the left and bottom borders. Here are the key points for solving the problem:

### Variables:
- Let $ x $ be the length of the left border (vertical).
- Let $ y $ be the length of the bottom border (horizontal).

### Costs:
- The cost of the left border is \$8 per foot, so total cost for the left border is $ 8x $.
- The cost of the bottom border is \$2 per foot, so total cost for the bottom border is $ 2y $.
- Total budget for fencing: \$560.

### Objective:
- Maximize the area of the right-angled triangular garden, which is $ A = \frac{xy}{2} $.
  
### Constraints:
- Total cost: $ 8x + 2y = 560 $.

### Steps to Solve:

1. **Express one variable in terms of the other**:
   From the cost constraint, $ 8x + 2y = 560 $, solve for $ y $:
   \[
   2y = 560 - 8x
   \]
   \[
   y = 280 - 4x
   \]

2. **Substitute into the area formula**:
   The area $ A $ is given by:
   \[
   A = \frac{xy}{2}
   \]
   Substituting $ y = 280 - 4x $:
   \[
   A = \frac{x(280 - 4x)}{2}
   \]
   Simplifying:
   \[
   A = \frac{280x - 4x^2}{2}
   \]
   \[
   A = 140x - 2x^2
   \]

3. **Maximize the area**:
   To find the maximum area, take the derivative of $ A $ with respect to $ x $ and set it equal to zero:
   \[
   \frac{dA}{dx} = 140 - 4x
   \]
   Setting the derivative equal to zero:
   \[
   140 - 4x = 0
   \]
   \[
   x = 35
   \]

4. **Find $ y $**:
   Substituting $ x = 35 $ into the equation $ y = 280 - 4x $:
   \[
   y = 280 - 4(35) = 280 - 140 = 140
   \]

5. **Calculate the maximum area**:
   The maximum area is:
   \[
   A = \frac{xy}{2} = \frac{35 \times 140}{2} = \frac{4900}{2} = 2450 \, \text{square feet}.
   \]

### Final Dimensions and Area:
- Left border $ x = 35 $ feet.
- Bottom border $ y = 140 $ feet.
- Maximum garden area = 2450 square feet.

Would you like more details or have any questions?

### Related Questions:
1. How would the problem change if the cost of the bottom fence increased to \$4 per foot?
2. What if the total budget was reduced to \$400? What would be the new dimensions?
3. How do you derive the maximum area formula for any budget constraint and different cost ratios?
4. Can you find the second derivative of the area function to confirm it's a maximum?
5. How does the optimal solution change if the shape is not a right triangle?

### Tip:
The vertex of a parabola (in this case, the area function) is where the function reaches its maximum or minimum, which is why setting the derivative to zero finds the optimal solution.

You are building a right-angled triangular flower garden along a stream. The fencing of the left border costs $8 per foot, while the fencing of the lower border costs $2 per foot. You want to spend $560 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose?

Learn how to maximize the area of a right-angled triangular garden while staying within a budget for fencing. The problem involves optimizing a quadratic function based on given cost constraints for the fencing of the left and bottom borders.

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