Question: Express the formula $P = 2l + 2w$ in terms of the width, $w$. Use the formula to find the width when the perimeter is $70$ and the length is $22$.
○ $w = \frac{P - l}{2}$; $w = 24$
○ $w = \frac{P}{2} - l$; $w = 13$
○ $w = P - 2l$; $w = 26$
○ $w = \frac{P - 2l}{2}$; $w = 9$
1. **State the problem:**
We are given the perimeter formula for a rectangle:
$$P = 2l + 2w$$
We need to express $w$ in terms of $P$ and $l$, then find $w$ when $P = 70$ and $l = 22$.
2. **Rearrange the formula to solve for $w$:**
Start with:
$$P = 2l + 2w$$
Subtract $2l$ from both sides:
$$P - 2l = 2w$$
Divide both sides by 2:
$$\frac{P - 2l}{\cancel{2}} = \frac{2w}{\cancel{2}}$$
which simplifies to:
$$w = \frac{P - 2l}{2}$$
3. **Substitute the given values $P=70$ and $l=22$ into the formula:**
$$w = \frac{70 - 2 \times 22}{2}$$
Calculate the numerator:
$$70 - 44 = 26$$
So:
$$w = \frac{26}{2}$$
Divide:
$$w = 13$$
4. **Conclusion:**
The correct formula for $w$ is:
$$w = \frac{P - 2l}{2}$$
and the width when $P=70$ and $l=22$ is:
$$w = 13$$
This matches the second option:
○ $w = \frac{P}{2} - l$; $w = 13$
Note that $\frac{P}{2} - l = \frac{70}{2} - 22 = 35 - 22 = 13$, which is algebraically equivalent to $\frac{P - 2l}{2}$.
Hence, the correct answer is the second option.