The room has an area of 176 square feet and is 5 feet longer than it is wide. Find the dimensions of the family room.

Question

The area of the room is 176 square feet and is five feet longer than its wide.

Answer 1

Hello Heidi, this is a great question. 

 

To solve problems like these, you need to translate the words into equations.

 

"The room has an area of 176 square feet." means

Area = 176

Area of a rectangle is also length times width which gives:

l * w = 176

 

"and is 5 feet longer than it is wide" means

length = width + 5

l = w + 5

 

Now that we have both equations we can use substitution to get that the area is:

w * (w + 5) = 176

 

Even though you've tagged this problem as completing the square, I think it makes more sense to solve it using factoring techniques. Distributing the w gives:

w² + 5w = 176

 

Rearranging to get the equation in standard form gives:

w² + 5w - 176 = 0

 

Now, 176 does not factor extremely easily. There are a lot of different factoring techniques you can use. When I divide 176 by 2, I see that 2 and 88 are factors. So I have that 2*8*11= 176 or 16*11=176. This pair meets the requirements of the problem where the room is 11 feet wide and 16 feet long. Looking at it mathematically:

(w + 16)(w - 11) = 0

The positive root of this equation is w = 11. Returning to the equation that length = width + 5 gives that length = 11 + 5, which returns 16 feet.