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 t ok buy 270 square feet of carpeting. What dimensions should the rug have?

Answer 1

Answer:

Dimension of rug is $(15\times 18)\ ft$

Step-by-step 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.

∴ Rug dimension will be $(23-2x)\times (26-2x)$

We know, area of rectangle= $width\times length$

Forming an equation for area of rug.

⇒$270= (23-2x)\times (26-2x)$

Now solving the equation to find the dimension of rug.

$270= (23-2x)\times (26-2x)$

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.

= $\frac{-(-98)\pm \sqrt{-98^{2}-4(4\times 328) } }{2\times 4}$

= $\frac{98\pm \sqrt{9604-4(1312) } }{8}$

Opening parenthesis.

= $\frac{98\pm \sqrt{9604-5248 } }{8}$

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

We know 66²=4356  and √a²=a  or -a

= $\frac{98\pm 66}{8}$

= $\frac{98+66}{8}\ or \ \frac{98-66}{8}$

= $20.5 \ or\ 4$

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

Ignoring decimal value, therefore taking value of x is 4 ft

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

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

⇒ Width of rug= $23- 2\times 4$

⇒ Width of rug= $23-8= 15\ ft$

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

⇒ Length of rug= $(26-2\times 4)$

⇒ Length of rug= $26-8= 18\ ft$

Hence, dimension of rug is $(15\times 18)\ ft$