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Which quadratic equation can be used to determine the thickness of the border, [tex]x[/tex], if the table and border must have an area of 3,276 square inches and the table is 36 inches wide and 72 inches long without the border?

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]

Sagot :

GinnyAnswer
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]
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