To express the formula [tex]\( P = 2l + 2w \)[/tex] in terms of the length [tex]\( l \)[/tex] and find the length when the perimeter [tex]\( P \)[/tex] is 68 and the width [tex]\( w \)[/tex] is 13, let's proceed step-by-step:
1. Express the formula in terms of [tex]\( l \)[/tex]:
[tex]\[
P = 2l + 2w
\][/tex]
To solve for [tex]\( l \)[/tex], we need to isolate [tex]\( l \)[/tex] on one side of the equation:
[tex]\[
P - 2w = 2l
\][/tex]
[tex]\[
l = \frac{P - 2w}{2}
\][/tex]
2. Substitute the given values for [tex]\( P \)[/tex] and [tex]\( w \)[/tex]:
Given [tex]\( P = 68 \)[/tex] and [tex]\( w = 13 \)[/tex]:
[tex]\[
l = \frac{68 - 2 \times 13}{2}
\][/tex]
Simplify within the parentheses:
[tex]\[
l = \frac{68 - 26}{2}
\][/tex]
Calculate the difference:
[tex]\[
l = \frac{42}{2}
\][/tex]
Divide by 2:
[tex]\[
l = 21
\][/tex]
3. Check the provided options:
Let's analyze each option and see where it fits:
- [tex]\( l = \frac{P}{2} - 2w \)[/tex]:
Substitution gives:
[tex]\[
l = \frac{68}{2} - 2 \times 13 = 34 - 26 = 8
\][/tex]
This value is incorrect.
- [tex]\( l = \frac{P - 200}{2} \)[/tex]:
Substitution gives:
[tex]\[
l = \frac{68 - 200}{2} = \frac{-132}{2} = -66
\][/tex]
This value is incorrect.
- [tex]\( t = \frac{P}{2} - w \)[/tex]:
Substitution gives:
[tex]\[
t = \frac{68}{2} - 13 = 34 - 13 = 21
\][/tex]
The calculated value of [tex]\( t \)[/tex] matches [tex]\( l = 21 \)[/tex]. So this option is correct, but we need to use [tex]\( l \)[/tex] instead of [tex]\( t \)[/tex].
- [tex]\( l = \frac{P - 40}{2} \)[/tex]:
Substitution gives:
[tex]\[
l = \frac{68 - 40}{2} = \frac{28}{2} = 14
\][/tex]
This value is incorrect.
Therefore, the correct length from the calculated values is [tex]\( 21 \)[/tex]. The correct transformed formula that matches the calculated length is [tex]\( t = \frac{P}{2} - w \)[/tex], which simplifies to [tex]\( l = 21 \)[/tex].
