Tabitha wants to hang a painting in a gallery. The painting and frame must have an area of 58 square feet. The painting is 7 feet wide by 8 feet long. Which quadratic equation can be used to determine the thickness of the frame, [tex]$x$[/tex]?

A. [tex]$x^2 + 15x - 2 = 0$[/tex]
B. [tex][tex]$x^2 + 15x + 58 = 0$[/tex][/tex]
C. [tex]$4x^2 + 30x - 2 = 0$[/tex]
D. [tex]$4x^2 + 30x + 58 = 0$[/tex]



Answer :

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]