Let's solve the problem step by step.
1. Define the Variables:
- Let [tex]\( w \)[/tex] be the width of the painting.
- The length [tex]\( l \)[/tex] is given as three more than twice the width: [tex]\( l = 2w + 3 \)[/tex].
2. Understand the Perimeter:
- The framing material used corresponds to the perimeter of the rectangle. The formula for the perimeter [tex]\( P \)[/tex] of a rectangle is:
[tex]\[ P = 2 \times ( \text{length} + \text{width}) \][/tex]
- Given that the perimeter is 30 inches, we can write:
[tex]\[ 2 \times (l + w) = 30 \][/tex]
3. Substitute the Length:
- Substitute [tex]\( l = 2w + 3 \)[/tex] into the perimeter equation:
[tex]\[ 2 \times ((2w + 3) + w) = 30 \][/tex]
4. Simplify the Equation:
- Combine like terms inside the parentheses:
[tex]\[ 2 \times (3w + 3) = 30 \][/tex]
- Distribute the 2:
[tex]\[ 6w + 6 = 30 \][/tex]
5. Solve for Width [tex]\( w \)[/tex]:
- Subtract 6 from both sides of the equation:
[tex]\[ 6w = 24 \][/tex]
- Divide by 6:
[tex]\[ w = 4 \][/tex]
6. Find the Length:
- Substitute the width [tex]\( w \)[/tex] back into the expression for the length:
[tex]\[ l = 2w + 3 \][/tex]
[tex]\[ l = 2(4) + 3 \][/tex]
[tex]\[ l = 8 + 3 \][/tex]
[tex]\[ l = 11 \][/tex]
So, the length of the painting is 11 inches.
