Cynthia Besch wants to buy a rug for a room that is 29 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 756 square feet of carpeting. What dimension should the rug have?

Given :

Length = 30 ft

Width = 29 ft

Area of carpeting = 756 sq ft

Let the uniform strip around floor is of length = x ft

This x feet will be 2 times along the length and

2 times along the width of room.

So, length of rug = 30 -2× x

Width of rug = 29 -2 × x

It will be rectangular in shape.

Area of a rectangle = Length × width 

756 = (30 - 2x )(29 -2x)

756 = 870 - 58x -60x + 4x²

4x² - 118x + 870 - 756 = 0

4x² - 118x + 114 = 0

Taking 2 common from above equation 

2(2x² - 59x + 57) = 0

2x² - 59x + 57 = 0

Now, by middle-term split method :

2x² - 57x - 2x + 57 = 0

x(2x - 57) -1(2x-57) = 0

(x-1) (2x- 57) = 0

So, x = 1, 57/2

So, possible dimensions are:

For x = 1

Length = 30 - 2(1) 

= 30 - 2 

= 28 feet

And  width = 29 - 2(1)

= 29 - 2

= 27 feet

For x = 57/2

Length = 30 -2(57/2)

= 30 - 57

= -27

Since, length cannot be in negative . So, only possible dimensions of the rug are 28 feet and 27 feet.