Certainly! Let's start by expressing the given formula [tex]\( P = 2l + 2w \)[/tex] in terms of the width [tex]\( w \)[/tex].
First, we isolate [tex]\( w \)[/tex] in the equation:
1. The given formula is:
[tex]\[ P = 2l + 2w \][/tex]
2. Subtract [tex]\( 2l \)[/tex] from both sides to isolate the terms involving [tex]\( w \)[/tex]:
[tex]\[ P - 2l = 2w \][/tex]
3. Divide both sides of the equation by 2 to solve for [tex]\( w \)[/tex]:
[tex]\[ w = \frac{P - 2l}{2} \][/tex]
Now, we use this formula to find the width [tex]\( w \)[/tex] when the perimeter [tex]\( P \)[/tex] is 70 and the length [tex]\( l \)[/tex] is 22:
1. Substitute [tex]\( P = 70 \)[/tex] and [tex]\( l = 22 \)[/tex] into the formula:
[tex]\[ w = \frac{70 - 2 \cdot 22}{2} \][/tex]
2. Calculate the expression inside the parentheses first:
[tex]\[ 70 - 2 \cdot 22 = 70 - 44 = 26 \][/tex]
3. Now divide 26 by 2:
[tex]\[ w = \frac{26}{2} = 13 \][/tex]
Therefore, the width [tex]\( w \)[/tex] is [tex]\( 13 \)[/tex].
This matches the fourth option given in the problem:
[tex]\[ w = \frac{P - 2l}{2} ; w = 13 \][/tex]
