To determine the quadratic equation that represents the thickness of the frame, follow these steps:
1. Identify the dimensions of the mirror:
- Width of the mirror ([tex]\( w \)[/tex]) = 3 feet
- Length of the mirror ([tex]\( l \)[/tex]) = 4 feet
2. Calculate the area of the mirror:
[tex]\[
\text{Area of the mirror} = w \times l = 3 \times 4 = 12 \text{ square feet}
\][/tex]
3. Express the total area including the frame:
- The frame adds a uniform thickness ([tex]\( x \)[/tex]) around the mirror.
- Hence, the overall dimensions including the frame would be:
- Width including the frame = [tex]\( w + 2x \)[/tex]
- Length including the frame = [tex]\( l + 2x \)[/tex]
4. Create an equation for the total area (mirror + frame):
[tex]\[
\text{Total area} = (\text{width including the frame}) \times (\text{length including the frame})
\][/tex]
Substituting the dimensions:
[tex]\[
15 = (3 + 2x) \times (4 + 2x)
\][/tex]
5. Expand the equation:
[tex]\[
15 = (3 + 2x)(4 + 2x)
\][/tex]
[tex]\[
15 = 12 + 6x + 8x + 4x^2
\][/tex]
[tex]\[
15 = 4x^2 + 14x + 12
\][/tex]
6. Form the quadratic equation:
[tex]\[
4x^2 + 14x + 12 = 15
\][/tex]
Subtract 15 from both sides to set the equation to zero:
[tex]\[
4x^2 + 14x + 12 - 15 = 0
\][/tex]
[tex]\[
4x^2 + 14x - 3 = 0
\][/tex]
Therefore, the quadratic equation that can be used to determine the thickness of the frame [tex]\( x \)[/tex] is:
[tex]\[
\boxed{4x^2 + 14x - 3 = 0}
\][/tex]
So, the correct option is:
[tex]\[
4x^2 + 14x - 3 = 0
\][/tex]
