Let's determine the quadratic equation that can be used to find the thickness of the border \( x \).
First, let's denote:
- The width of the table, \( w = 36 \) inches,
- The length of the table, \( l = 72 \) inches,
- The total area of the table and border combined, \( A_{\text{total}} = 3,276 \) square inches.
The area of the table without the border is given by:
[tex]\[ A_{\text{table}} = w \times l = 36 \times 72 = 2,592 \text{ square inches} \][/tex]
To find the thickness of the border, \( x \), we need to consider the dimensions of the table including the border. The width including the border will be \( 36 + 2x \) and the length including the border will be \( 72 + 2x \).
The area that includes both the table and the border can be expressed as:
[tex]\[ (36 + 2x)(72 + 2x) = A_{\text{total}} \][/tex]
Expanding the left-hand side:
[tex]\[ (36 + 2x)(72 + 2x) = 36 \times 72 + 2x \times 72 + 2x \times 36 + 4x^2 \][/tex]
[tex]\[ = 2,592 + 144x + 72x + 4x^2 \][/tex]
[tex]\[ = 2,592 + 216x + 4x^2 \][/tex]
Since this area must equal the total area, we set it equal to 3,276:
[tex]\[ 2,592 + 216x + 4x^2 = 3,276 \][/tex]
Rearranging to form a quadratic equation:
[tex]\[ 4x^2 + 216x + 2,592 - 3,276 = 0 \][/tex]
[tex]\[ 4x^2 + 216x - 684 = 0 \][/tex]
Therefore, the quadratic equation that can be used to determine the thickness of the border \( x \) is:
[tex]\[ 4x^2 + 216x - 684 = 0 \][/tex]
