Sure! Let's go through the problem step-by-step.
### Step 1: Express the formula in terms of [tex]\( l \)[/tex]
We start with the given formula for the perimeter:
[tex]\[ P = 2l + 2w \][/tex]
We need to express this formula in terms of [tex]\( l \)[/tex]. To do so, we will solve for [tex]\( l \)[/tex]:
1. Subtract [tex]\( 2w \)[/tex] from both sides of the equation:
[tex]\[ P - 2w = 2l \][/tex]
2. Divide both sides by 2 to isolate [tex]\( l \)[/tex]:
[tex]\[ l = \frac{P - 2w}{2} \][/tex]
### Step 2: Substitute the given values into the new formula
We are given that the perimeter [tex]\( P \)[/tex] is 68 and the width [tex]\( w \)[/tex] is 13. Now we substitute these values into our new formula:
[tex]\[ l = \frac{68 - 2 \cdot 13}{2} \][/tex]
### Step 3: Simplify the expression
First, calculate the term inside the parentheses:
[tex]\[ 2 \cdot 13 = 26 \][/tex]
Now substitute this back into the equation:
[tex]\[ l = \frac{68 - 26}{2} \][/tex]
Next, perform the subtraction:
[tex]\[ 68 - 26 = 42 \][/tex]
Now substitute this result back into the equation:
[tex]\[ l = \frac{42}{2} \][/tex]
Finally, perform the division:
[tex]\[ l = 21 \][/tex]
### Step 4: Conclusion
The length [tex]\( l \)[/tex] when the perimeter is 68 and the width is 13 is 21.
So, the length [tex]\( l = 21 \)[/tex] when the perimeter [tex]\( P = 68 \)[/tex] and the width [tex]\( w = 13 \)[/tex].
