Given the problem, let's solve it step-by-step.
Katie has 60 inches of material available to create a rectangular frame. The length of the frame is supposed to be 3 inches more than twice the width. We need to determine the largest possible length of the frame under these conditions.
### Step-by-Step Solution:
1. Define the variables:
- Let [tex]\( w \)[/tex] represent the width of the rectangular frame.
- The length of the rectangular frame [tex]\( l \)[/tex] is given by [tex]\( l = 2w + 3 \)[/tex].
2. Use the information about the perimeter:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by [tex]\( P = 2l + 2w \)[/tex].
- Given that Katie has 60 inches of material, [tex]\( P = 60 \)[/tex].
3. Write the perimeter equation and substitute the given length:
- Substitute the expression for [tex]\( l \)[/tex] into the perimeter equation:
[tex]\[
2(2w + 3) + 2w = 60
\][/tex]
4. Solve the equation step-by-step:
- Distribute the multiplication:
[tex]\[
4w + 6 + 2w = 60
\][/tex]
- Combine like terms:
[tex]\[
6w + 6 = 60
\][/tex]
- Subtract 6 from both sides to isolate the term with [tex]\( w \)[/tex]:
[tex]\[
6w = 54
\][/tex]
- Divide by 6 to solve for [tex]\( w \)[/tex]:
[tex]\[
w = 9
\][/tex]
5. Calculate the length [tex]\( l \)[/tex] using the width [tex]\( w \)[/tex]:
- Substitute [tex]\( w = 9 \)[/tex] back into the length equation:
[tex]\[
l = 2(9) + 3 = 18 + 3 = 21
\][/tex]
### Conclusion:
The largest possible length of the rectangular frame that Katie can make, given 60 inches of material and her constraints, is 21 inches.
Thus, the solution is:
[tex]\[
\boxed{21 \text{ inches}}
\][/tex]
