Sure, let's work through this problem step by step.
1. Identify the dimensions of the painting:
- Width = 7 feet
- Length = 8 feet
2. Calculate the area of the painting:
[tex]\[
\text{Area of the painting} = \text{Width} \times \text{Length} = 7 \times 8 = 56 \text{ square feet}
\][/tex]
3. Identify the total area including the frame:
[tex]\[
\text{Total area (painting + frame)} = 58 \text{ square feet}
\][/tex]
4. Define the thickness of the frame as [tex]\(x\)[/tex]:
5. Express the dimensions of the painting with the frame:
- Width including the frame: [tex]\(7 + 2x\)[/tex] (since the frame adds thickness on both sides)
- Length including the frame: [tex]\(8 + 2x\)[/tex]
6. Write the equation for the total area including the frame:
[tex]\[
\text{Total area} = (\text{Width including frame}) \times (\text{Length including frame})
\][/tex]
[tex]\[
58 = (7 + 2x) \times (8 + 2x)
\][/tex]
7. Expand the equation:
[tex]\[
58 = (7 + 2x)(8 + 2x)
\][/tex]
Using the distributive property:
[tex]\[
(7 + 2x)(8 + 2x) = 7 \times 8 + 7 \times 2x + 2x \times 8 + 2x \times 2x
\][/tex]
[tex]\[
= 56 + 14x + 16x + 4x^2
\][/tex]
[tex]\[
= 56 + 30x + 4x^2
\][/tex]
8. Set up the quadratic equation:
[tex]\[
58 = 56 + 30x + 4x^2
\][/tex]
9. Rearrange the equation to set it to zero:
[tex]\[
4x^2 + 30x + 56 - 58 = 0
\][/tex]
[tex]\[
4x^2 + 30x - 2 = 0
\][/tex]
So, the correct quadratic equation that can be used to determine the thickness of the frame [tex]\(x\)[/tex] is:
[tex]\[
4 x^2 + 30 x - 2 = 0
\][/tex]
Therefore, the correct answer is:
[tex]\[
4x^2 + 30x - 2 = 0
\][/tex]
