Certainly! Let's walk through the problem step-by-step.
We start with the formula for the perimeter [tex]\( P \)[/tex] of a rectangle, given in terms of its length [tex]\( l \)[/tex] and width [tex]\( w \)[/tex]:
[tex]\[ P = 2l + 2w \][/tex]
We need to express this formula in terms of the length [tex]\( l \)[/tex]. To do this, we'll solve for [tex]\( l \)[/tex].
First, we isolate the terms involving [tex]\( l \)[/tex]:
[tex]\[ P = 2l + 2w \][/tex]
Subtract [tex]\( 2w \)[/tex] from both sides:
[tex]\[ P - 2w = 2l \][/tex]
Next, divide both sides by 2 to solve for [tex]\( l \)[/tex]:
[tex]\[ l = \frac{P - 2w}{2} \][/tex]
Now we have the formula in terms of the length [tex]\( l \)[/tex]:
[tex]\[ l = \frac{P - 2w}{2} \][/tex]
We can use this formula to find the length [tex]\( l \)[/tex] when the perimeter [tex]\( P \)[/tex] is 68 and the width [tex]\( w \)[/tex] is 13.
Substitute [tex]\( P = 68 \)[/tex] and [tex]\( w = 13 \)[/tex] into the formula:
[tex]\[ l = \frac{68 - 2 \cdot 13}{2} \][/tex]
Calculate within the parentheses first:
[tex]\[ l = \frac{68 - 26}{2} \][/tex]
Simplify the subtraction:
[tex]\[ l = \frac{42}{2} \][/tex]
Finally, divide:
[tex]\[ l = 21 \][/tex]
So, the length [tex]\( l \)[/tex] is 21. This matches the given choice:
[tex]\[ l = \frac{P}{2} - w ; l = 21 \][/tex]
Thus, the correct option is:
[tex]\[ l = \frac{P}{2} - w ; l = 21 \][/tex]
