Catarina and Tom want to buy a rug for a room that is 11 by 20 feet. They want to leave an even strip of flooring uncovered around the edges of the room. How wide a strip will they have if they buy a rug with an area of 112 square feet? They will have a \( \square \mathrm{ft} \) wide strip of flooring uncovered.

To find the width of the strip of flooring that will remain uncovered, we first need to determine the dimensions of the rug and the total area of the room. The room is 11 by 20 feet, giving a total area of: \[ 11 \times 20 = 220 \text{ square feet} \] Catarina and Tom want a rug with an area of 112 square feet. The area of the uncovered strip can be found by subtracting the area of the rug from the area of the room: \[ 220 \text{ square feet} - 112 \text{ square feet} = 108 \text{ square feet} \] Now, if we let \( x \) be the width of the strip on each side, the dimensions of the rug can be represented as: Width of rug = \( 11 - 2x \) Length of rug = \( 20 - 2x \) Hence, the area of the rug is: \[ (11 - 2x)(20 - 2x) = 112 \] Now, we can set up the equation: \[ 220 - 62x + 4x^2 = 112 \] Rearranging gives: \[ 4x^2 - 62x + 108 = 0 \] This can be solved using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plugging in \( a = 4 \), \( b = -62 \), and \( c = 108 \): \[ x = \frac{62 \pm \sqrt{(-62)^2 - 4 \cdot 4 \cdot 108}}{2 \cdot 4} \] Calculating further leads to: \[ x = \frac{62 \pm \sqrt{3844 - 1728}}{8} \] \[ x = \frac{62 \pm \sqrt{2116}}{8} \] \[ x = \frac{62 \pm 46}{8} \] This gives two possible solutions for \( x \): 1. \( x = \frac{108}{8} = 13.5 \) (not possible since it's wider than the room) 2. \( x = \frac{16}{8} = 2\) Thus, the width of the strip of flooring uncovered will be \( \boxed{2} \) feet.