Part 1 of 2 A carpenter is building a rectangular room with a fixed perimeter of 360 ft . What dimensions would yield the maximum area Length \( (\mathrm{ft})=\square \)

To maximize the area of a rectangular room with a fixed perimeter, we should remember that a square shape yields the largest area for a given perimeter. Given a perimeter of 360 ft, we can find the length of each side of the square. Since the perimeter \( P = 4L \) (where \( L \) is the length of each side), we can rearrange this to find \( L = \frac{P}{4} = \frac{360}{4} = 90 \) ft. Thus, the maximum area can be achieved with dimensions of 90 ft by 90 ft. Now let's think about the experience of building! Imagine your carpenter friend, standing in a sunny yard with all their tools laid out, knowing that crafting something with precise dimensions will not only optimize space but also become a cozy refuge. It's one of those moments that blends art and mathematics seamlessly. Cheers to well-thought-out spaces!