To determine the thickness of the border [tex]\( x \)[/tex], let's set up the equation based on the given information:
1. Table Dimensions:
- Width of the table: 36 inches
- Length of the table: 72 inches
2. Area of the Table (without border):
[tex]\[
\text{Area of the table} = \text{Width} \times \text{Length} = 36 \times 72 = 2592 \, \text{square inches}
\][/tex]
3. Total Area (including the border):
- Given total area: 3276 square inches
4. Equation Setup:
Let's denote the thickness of the border as [tex]\( x \)[/tex]. The total dimensions of the table including the border will be:
- New Width: [tex]\( 36 + 2x \)[/tex]
- New Length: [tex]\( 72 + 2x \)[/tex]
Next, let's calculate the total area including the border:
[tex]\[
(\text{New Width}) \times (\text{New Length}) = (36 + 2x) \times (72 + 2x)
\][/tex]
5. Equation Formation:
According to the problem, the total area is given by:
[tex]\[
(36 + 2x)(72 + 2x) = 3276
\][/tex]
6. Expand the Equation:
[tex]\[
36 \times 72 + 36 \times 2x + 72 \times 2x + 4x^2 = 3276
\][/tex]
[tex]\[
2592 + 72x + 144x + 4x^2 = 3276
\][/tex]
[tex]\[
2592 + 216x + 4x^2 = 3276
\][/tex]
7. Rearrange into Standard Quadratic Form:
[tex]\[
4x^2 + 216x + 2592 = 3276
\][/tex]
[tex]\[
4x^2 + 216x + 2592 - 3276 = 0
\][/tex]
[tex]\[
4x^2 + 216x - 684 = 0
\][/tex]
So, the correct quadratic equation that can be used to determine the thickness of the border [tex]\( x \)[/tex] is:
[tex]\[
4x^2 + 216x - 684 = 0
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{4x^2 + 216x - 684 = 0}
\][/tex]
