A rectangular room containing 198 square feet is 7 feet longer than it is wide. How long of a piece of crown molding is needed to trim the perimeter of the ceiling?

L = W + 7

198 = LW = W(W+7)

Solve for W then L.

Perimeter = 2W + 2L

To find the length of the crown molding needed to trim the perimeter of the ceiling, we first need to determine the dimensions of the rectangular room.

Let's assume the width of the room is "x" feet. According to the problem, the length of the room is 7 feet longer than the width, which means the length is "x + 7" feet.

We know that the area of a rectangle is given by the formula: Area = Length × Width.

Given that the area of the room is 198 square feet, we can set up the following equation:

198 = (x + 7) × x

Expanding the equation:

198 = x^2 + 7x

Rearranging the equation:

x^2 + 7x - 198 = 0

Now, we can solve this quadratic equation to find the width (x) of the room.

Using factoring, completing the square, or the quadratic formula, you'll find that the possible solutions for "x" are -22 and 9. However, since we are dealing with dimensions, the width of the room cannot be negative. Therefore, x = 9.

So, the width of the room is 9 feet, and the length of the room is x + 7 = 9 + 7 = 16 feet.

To find the perimeter of the room, we add up all the sides:

Perimeter = 2 × (Length + Width)

Plugging in the values:

Perimeter = 2 × (16 + 9)
Perimeter = 50 feet

Finally, to determine the length of the crown molding needed to trim the perimeter of the ceiling, we know that crown molding is placed around the top perimeter of the room. Therefore, the length of the crown molding needed is equal to the perimeter of the room, which is 50 feet.