A carpenter is building a rectangular room with a fixed perimeter of $180$ ft $.$ What are the dimensions of the largest room that can be built? What is its area?

$45$ ft by $135ft$;$6,075f{t}^2$

$45$ ft by $45ft$;$2,025f{t}^2$

$90$ ft by $90ft$;$8,100f{t}^2$

$18$ ft by $162ft$;$2,916f{t}^2$