An oriental rug is 3 feet longer than it is wide. If the diagonal of the rug is 12 feet, find its dimensions to the nearest tenth of a foot.

Question

Finding the dimensions of the rug.

Answer 1

Let w represent the width (in feet) of the rug. Its length is then 3 + w. A length, width, and diagonal of a rectangle make a right triangle (with the diagonal as the hypotenuse), so since the diagonal is 12, the Pythagorean theorem gives 

 

w^2 + (3 + w)^2 = 12^2

 

w^2 + 9 + 6w + w^2 = 144

 

2w^2 + 6w - 135 = 0

 

By the quadratic formula, w = (-3 +/- √31)/2. Since width is positive, w = (-3 + √31)/2, which is approximately 6.9. The dimensions of the rug are therefore 3 + 6.9 = 9.9 feet long by 6.9 feet wide.

Answer 2

I am going to assume the rug is rectangular.

Let w = the width

Let w + 3 = the length   Draw a rectangle and label the width, length, and diagonal. Then you can use the Pythagorean Theorem.

w² + (w+3)² = 12²

w² + w² + 6w + 9 = 144

2w² + 6w - 135 = 0     Since this won't factor, use the quadratic formula.

w = [-6 ±√(6² -4(2)(-135)]/2(2)

w = (-6 ±√1116)/4)

w = (-6 ± 6√31)/4

w = (-3 ± 3√31)/2

w = (-3 ± 16.703)/2

w = 6.9 feet

w+3 =9.9 feet