contestada

Cynthia Besch wants to buy a rug for a room that is 20 ft wide and 26 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 432 square feet of carpeting. What dimensions should the rug​ have?

Respuesta :

altavistard
Let x represent the width of the "uniform strip."

Write formulas:  one for the length of the rug, one for the width.

If the room is 20 ft wide, the rug width would have to be 20-2x, and the run length would be 26-2x.

Rug area is 432 sq ft, and this equals (rug length)(rug width), or

432 = (26-2x)(20-2x)  = 520 -40x  - 52x  + 4x^2   and this is 432.

Subtr. 432 from both sides:    520 - 432 - 92x + 4x^2 = 0.  This is a quadratic equation that could be solved in various ways.

4x^2 - 92x + 88 = 0.  Let's reduce this by div. all terms by 4:

x^2 -23x + 22 = 0

Easily factored!  Note that -1x and -22x add up to -23x, and that (-1)(-12) = 22.  Thus, the factors are (x-1)(x-22) = 0, so that x = 1 or x = 22.
Remembering that x represents the strip width, we eliminate x = 22 and keep x = 1.

The rug dimensions should be (26-2) by (20-2), or 24 by 18 feet.

Check:  Does this area come out to 432 sq ft?   (24)(18) = 432/  YES
onyebuchinnaji

The dimensions of the rug should be 18 feet and 24 feet

The given parameters;

dimension of the room = 20 ft by 26 ft

maximum area of rug she can afford = 432 ft²

To find:

  • the dimension of the rug

If they will be a uniform stripe of floor around the rug, then let the uniform excess length of the floor to removed from each dimension = y

[tex]Area = Length \times width\\\\432 = (20 - y ) \times (26-y)\\\\432 = 520 -20y -26y + y^2\\\\y^2 - 46y + 88 = 0\\\\\\solve \ the \ quadratic \ equation \ using \ formula \ method;\\\\a = 1, \ b = - 46, \ c = 88\\\\y = \frac{-b \ \ +/- \ \ \sqrt{b^2 -4ac} }{2a} \\\\y = \frac{-(-46) \ \ +/- \ \ \sqrt{(-46)^2 -4(1\times 88)} }{2(1)} \\\\y = 44 \ \ \ or \ \ \ 2[/tex]

The value of y cannot be greater than any of the original dimension, we will choose y = 2

The uniform dimension of the floor to be covered by the maximum area of rug she can afford = (20 - 2) and (26 - 2) = 18 and 24

Thus, the dimensions of the rug should be 18 feet and 24 feet

Learn more here: https://brainly.com/question/12932084

Otras preguntas