Question

Cynthia Besch wants to buy a rug for a room that is 24 ft wide and 33 ft long. She

wants to leave a uniform strip of floor around the rug. She can afford to buy 400
square feet of carpeting. What dimensions should the rug have?

Answer 1

Answer: 16 ft by 25 ft

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Step-by-step explanation:

Refer to the diagram below.

The outer dimensions 24 ft and 33 ft shrink down to 24-2x ft and 33-2x ft respectively. This subtraction of 2x is due to subtracting two copies of x per side.

The carpet has area of (24-2x)(33-2x)

Cynthia can afford to buy 400 sq ft of carpet

So we set 400 equal to that previous expression and solve for x

(24-2x)(33-2x) = 400

24(33-2x) - 2x(33-2x) = 400

792-48x - 66x + 4x^2 = 400

4x^2 - 114x + 792 = 400

4x^2 - 114x + 792-400 = 0

4x^2 - 114x + 392 = 0

From here, use the quadratic formula to isolate x.

Plug in a = 4, b = -114, c = 392

`x = (-b\pm√(b^2-4ac))/(2a)\\\\x = (-(-114)\pm√((-114)^2-4(4)(392)))/(2(4))\\\\x = (114\pm√(6724))/(8)\\\\x = (114\pm82)/(8)\\\\x = (114+82)/(8) \ \text{ or } \ x = (114-82)/(8)\\\\x = (196)/(8) \ \text{ or } \ x = (32)/(8)\\\\x = 24.5 \ \text{ or } \ x = 4\\\\`

The two possible solutions are x = 24.5 and x = 4

But if x = 24.5, then 24 - 2x = 24 - 2*24.5 = -25 which isn't possible. We cannot have a negative width or negative length for the carpet.

Luckily x = 4 does work since

24 - 2x = 24 - 2*4 = 16

33 - 2x = 33 - 2*4 = 25

Both results are positive.

Therefore, the carpet has dimensions of 16 ft by 25 ft

Check: 16*25 = 400, so the answer is confirmed