Ken wants to build a table and put a border around it. The table and border must have an area of 3,276 square inches. The table is 36 inches wide and 72 inches long without the border. Which quadratic equation can be used to determine the thickness of the border, [tex]$x$[/tex]?

A. [tex]$4x^2 + 216x + 2,592 = 0$[/tex]
B. [tex]$4x^2 + 216x - 684 = 0$[/tex]
C. [tex]$2x^2 + 216x - 3,276 = 0$[/tex]
D. [tex]$x^2 + 108x + 3,276 = 0$[/tex]



Answer :

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]