Cynthia Besch wants to buy a rug for a room that is 23 ft wide and 29 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 475 square feet of carpeting. What dimensions should the rug​ have?

Ur papa

Well, if Cynthia wants to leave a uniform strip of floor around the rug, it's only fair that the rug has some breathing room. Let's try not to suffocate it with the walls.

Now, since Cynthia can afford to buy 475 square feet of carpeting, we can use that information to find the dimensions of the rug. If we subtract the area of the rug from the total area she can afford, we'll get the area of the strip. It's like a game of Rug Hide-and-Seek!

Let's solve this mystery step by step. The area of the room is calculated by multiplying width by length: 23 ft × 29 ft = 667 sq. ft.

If we subtract the area of the rug from the total area of the room, we get the area of the strip: 667 sq. ft - rug area = strip area.

Since Cynthia wants to leave a uniform strip, let's assume the strip has the same width all around the rug. So, the width of the strip will be the same as the width of the rug, plus twice the width of the strip.

Now, we need to express the area of the strip in terms of width.

The width of the strip = width of the rug + 2 × width of the strip.

Rearranging the equation, we get:

Width of the strip = (width of the room - width of the rug) / 2.

Let's substitute the values we know and try to crack this rug mystery. Oh, but watch out for any rug thieves along the way!

Width of the strip = (23 ft - width of the rug) / 2.

Now, we can express the area of the strip in terms of the width of the rug:

Area of the strip = width of the strip × length of the strip.

Substituting the values we know, we get:

475 sq. ft = [(23 ft - width of the rug) / 2] × 29 ft.

Now, we have an equation to solve. Let's gather our clown detective magnifying glasses and solve the mystery of the rug dimensions:

475 sq. ft = [(23 ft - width of the rug) × 29 ft] / 2.

Now, let's solve for the width of the rug and find out what kind of secrets it's been hiding:

Width of the rug = (23 ft × 29 ft - 950 sq. ft) / (29 ft / 2).

Simplifying this equation and unveiling the rug's true identity, we find:

Width of the rug = 11 ft.

So, the rug should have a width of 11 ft, leaving a uniform strip around it. Now, it's time to buy a rug and give that room some cozy, clown-approved flair!

To find the dimensions of the rug, we can follow these steps:

1. Calculate the area of the room: The area of the room can be found by multiplying the width and length. In this case, the width is 23 ft and the length is 29 ft. The calculation would be: Area = Width × Length.

Area of the room = 23 ft × 29 ft = 667 ft².

2. Calculate the area that will be covered by the rug: Since Cynthia wants to leave a uniform strip of floor around the rug, we need to subtract the area of the strip from the total area of the room. Let's assume the width of the strip is "x" feet.

The width of the rug will be: (23 ft - 2x) ft.
The length of the rug will be: (29 ft - 2x) ft.

The area covered by the rug can be calculated as: Area of the rug = Width of the rug × Length of the rug.

3. Set up an equation: We know that the area covered by the rug should be equal to the amount of carpeting Cynthia can afford, which is 475 ft². So, we can set up the equation:

Area of the rug = 475 ft².

(23 ft - 2x) ft × (29 ft - 2x) ft = 475 ft².

4. Solve the equation: Multiply the terms to simplify the equation:

(23 × 29) ft² - 2x(23 + 29) ft² + 4x² ft² = 475 ft².

667 ft² - 106x ft² + 4x² ft² = 475 ft².

Rearranging the equation:

4x² - 106x + (667 - 475) = 0.

4x² - 106x + 192 = 0.

Now, you can solve this quadratic equation using factoring, completing the square, or the quadratic formula.

5. Once you find the values of x (the width of the strip) that satisfy the equation, substitute them back into the expressions for the width and length of the rug to find the actual dimensions.

Remember to check if these dimensions fit within the room's dimensions.

23 - 2 w = width

29 - 2 w = length
(23-2w)(29-2w) =475
solve quadratic for w
https://www.mathsisfun.com/quadratic-equation-solver.html
then use w to get width and length