Question

Cynthia besch wants to buy a rug for a room that is 23ft wide and 26ft long. She wants t ok leave a uniform strip of floor around the rug. She can only afford to buy 270 square feet of carpeting. What dimensions should the rug have?

Answer 1

`(15* 18)\ ft`

Explanation:

Given: Area of rug= 270 ft²

Cynthia besch wants to buy a rug for a room that is 23ft wide and 26ft long.

Let´s assume the uniform strip size of floor around rug be "x".

As given, Cynthia wants to leave a uniform strip of floor around the rug.

Considering the shape of rug as rectangle.

∴ Rug dimension will be
`(23-2x)* (26-2x)`

We know, area of rectangle=
`width* length`

Forming an equation for area of rug.

⇒
`270= (23-2x)* (26-2x)`

Now solving the equation to find the dimension of rug.

`270= (23-2x)* (26-2x)`

Using distributive property of multiplication.

⇒
`270= 598-46x-52x+4x^(2)`

⇒
`598-98x+4x^(2)= 270`

Subtracting both side by 270

⇒
`4x^(2)-98x+328= 0`

using quadratic formula to solve the equation.

Formula:
`\frac{-b\pm \sqrt{b^(2)-4(ac) } }{2a}`

∴ In the expression , we have a= 4, b= -98 and c= 328.

Now, subtituting the value in the formula.

=
`\frac{-(-98)\pm \sqrt{-98^(2)-4(4* 328) } }{2* 4}`

=
`(98\pm √(9604-4(1312) ) )/(8)`

Opening parenthesis.

=
`(98\pm √(9604-5248 ) )/(8)`

= \frac{98\pm \sqrt{4356}}{8}

We know 66²=4356 and √a²=a or -a

=
`(98\pm 66)/(8)`

=
`(98+66)/(8)\ or \ (98-66)/(8)`

=
`20.5 \ or\ 4`

∴ Value of x will be either 20.5 ft or 4 ft

Ignoring decimal value, therefore taking value of x as 4 ft.

Subtituting the value of x to find the dimension of rug.

Width of rug=
`(23-2x)`

⇒ Width of rug=
`23- 2* 4`

⇒ Width of rug=
`23-8= 15\ ft`

Next, Length of rug=
`(26-2x)`

⇒ Length of rug=
`26-2* 4`

⇒ Length of rug=
`26-8= 18\ ft`

Hence, dimension of rug is
`(15* 18)\ ft`