To find the dimensions of the rectangular flower garden that minimize the amount of fencing while enclosing a flower bed with an area of 36 square feet, we can use optimization techniques.

Step-by-Step Approach:

1. Define Variables: Let:

   • $l$ be the length of the rectangular garden (in feet),
   • $w$ be the width of the garden (in feet).

   The area constraint is: $l \times w = 36 \text{ square feet}$

2. Perimeter of the Garden: The amount of fencing needed is based on the perimeter of the rectangle: $P = 2l + 2w$

3. Substitute the Area Constraint: From the area constraint $l \times w = 36$, solve for $w$: $w = \frac{36}{l}$ Substituting this into the perimeter equation: $P = 2l + 2\left(\frac{36}{l}\right)$ Simplifying: $P = 2l + \frac{72}{l}$

4. Optimize the Perimeter: To minimize the perimeter, we take the derivative of $P$ with respect to $l$ and set it to zero: $\frac{dP}{dl} = 2 - \frac{72}{l^2} = 0$ Solving for $l$: $2 = \frac{72}{l^2}$ $l^2 = 36 \quad \Rightarrow \quad l = 6$

   Using $l = 6$ in the area equation to find $w$: $w = \frac{36}{6} = 6$

Thus, the garden is a square with dimensions $6 \times 6$ feet, requiring the least amount of fence.

Total Perimeter and Fencing Cost:

The total perimeter is: $P = 2(6) + 2(6) = 24 \text{ feet}$

The cost per foot of fencing is $3.99, plus an 8% sales tax. The total cost can be calculated as follows: $\text{Cost before tax} = 24 \times 3.99 = 95.76 \text{ dollars}$ $\text{Tax} = 0.08 \times 95.76 = 7.66 \text{ dollars}$ $\text{Total cost} = 95.76 + 7.66 = 103.42 \text{ dollars}$

Summary Table:

Parameter	Value
Dimensions of Garden	$6 \times 6$ feet
Total Perimeter	24 feet
Cost per Foot of Fence	$3.99
Total Cost before Tax	$95.76
Sales Tax (8%)	$7.66
Total Cost	$103.42

Would you like to explore any specific details of this solution?

Here are 5 related questions to consider:

1. How does the optimization process change if the area of the garden is different?
2. What if Eve decides to use a different type of fence, with a different cost per foot?
3. How would the tax rate affect the total cost if it were increased or decreased?
4. Can the dimensions be non-integer values, or must they always be whole numbers?
5. How does the shape of the garden (square vs. rectangular) affect the cost?

Tip: Always check if the minimum perimeter occurs when the shape is a square when dealing with rectangular enclosures!