Question

Solve the problem. A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 12 feet by 22 feet and the area of the rug is 75 square feet, how wide will the border be?

Answer 1

To find the border width around a rug in a 12 by 22 feet room where the rug area is 75 square feet, a quadratic equation is set up and solved for the width variable, representing the border width.

Step-by-step explanation:

The student is asking to find the width of the border around a rug that will fit in a 12 feet by 22 feet room, leaving an even border on all sides when the rug has an area of 75 square feet. To solve this, let's denote the border width as x. The dimensions of the rug will then be (12 - 2x) feet by (22 - 2x) feet. Since the area of the rug is given as 75 square feet, we can write the following equation:

(12 - 2x)(22 - 2x) = 75

Expanding the brackets gives us:

264 - 44x - 24x + 4x² = 75

Simplifying the equation, we get:

4x² - 68x + 189 = 0

Now we can solve this quadratic equation to find the value of x. The solutions to the equation are the possible border widths. However, we need to consider only the positive solution that will be viable considering the dimensions of the room.

Using a quadratic formula or factoring, we find the positive value of x to be the width of the border. This is how we arrive at the desired answer.