Certainly! Let's derive a formula for the width [tex]\( w \)[/tex] in terms of the perimeter [tex]\( P \)[/tex] and the length [tex]\( l \)[/tex] given the expression [tex]\( P = 2l + 2w \)[/tex].
1. Start with the perimeter formula:
[tex]\[ P = 2l + 2w \][/tex]
2. Isolate the terms involving width:
Subtract [tex]\( 2l \)[/tex] from both sides:
[tex]\[ P - 2l = 2w \][/tex]
3. Solve for width [tex]\( w \)[/tex]:
Divide both sides by 2:
[tex]\[ w = \frac{P - 2l}{2} \][/tex]
Now, using the perimeter [tex]\( P = 70 \)[/tex] and the length [tex]\( l = 22 \)[/tex], we can find the width using the derived formula.
[tex]\[ w = \frac{70 - 2 \cdot 22}{2} \][/tex]
Calculate the expression step-by-step:
1. Evaluate the multiplication inside the parentheses:
[tex]\[ 2 \cdot 22 = 44 \][/tex]
2. Subtract this value from the perimeter:
[tex]\[ 70 - 44 = 26 \][/tex]
3. Finally, divide by 2:
[tex]\[ \frac{26}{2} = 13 \][/tex]
Thus, the width [tex]\( w \)[/tex] is 13.
Let's cross-check this with the given options:
- [tex]\( w = \frac{P - l}{2} \)[/tex]
Substituting [tex]\( P = 70 \)[/tex] and [tex]\( l = 22 \)[/tex]:
[tex]\[ w = \frac{70 - 22}{2} = \frac{48}{2} = 24 \][/tex]
This is incorrect.
- [tex]\( w = \frac{P}{2} - l \)[/tex]
Substituting [tex]\( P = 70 \)[/tex] and [tex]\( l = 22 \)[/tex]:
[tex]\[ w = \frac{70}{2} - 22 = 35 - 22 = 13 \][/tex]
This is correct!
- [tex]\( w = \frac{P - 2l}{2} \)[/tex]
Substituting [tex]\( P = 70 \)[/tex] and [tex]\( l = 22 \)[/tex]:
[tex]\[ w = \frac{70 - 2 \cdot 22}{2} = \frac{70 - 44}{2} = \frac{26}{2} = 13 \][/tex]
This is also correct but not given in the same form as our derived formula.
- [tex]\( w = P - 2l \)[/tex]
Substituting [tex]\( P = 70 \)[/tex] and [tex]\( l = 22 \)[/tex]:
[tex]\[ w = 70 - 2 \cdot 22 = 70 - 44 = 26 \][/tex]
This is incorrect.
Based on the evaluation, the correct formula and corresponding value are:
[tex]\[ w = \frac{P}{2} - l \quad \Rightarrow \quad w = 13 \][/tex]
