Make a table. Eve is planning a rectangualr flower garden that will contain a 36-square-foot flower bed. She wants to enclose the garden with a short decorative fence that is sold in 1-foot length costs P3.99, plus 8% sales tax. Find the dimensions of the garden that requires the least amount of fence. 

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!

Eve is planning a rectangular flower garden that will contain a 36-square-foot flower bed. She wants to enclose the garden with a short decorative fence that is sold in 1-foot length costs P3.99, plus 8% sales tax. Find the dimensions of the garden that requires the least amount of fence.

Solve an optimization problem involving a rectangular flower garden with a 36-square-foot area. Learn how to minimize the perimeter of the garden to reduce fencing costs using calculus. Includes a step-by-step solution for optimizing the dimensions of the garden and calculating the total cost of the fence. Suitable for Grades 10-12.

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