Answer :
Answer:
Dimension of rug is [tex](15\times 18)\ ft[/tex]
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 [tex](23-2x)\times (26-2x)[/tex]
We know, area of rectangle= [tex]width\times length[/tex]
Forming an equation for area of rug.
⇒[tex]270= (23-2x)\times (26-2x)[/tex]
Now solving the equation to find the dimension of rug.
[tex]270= (23-2x)\times (26-2x)[/tex]
Using distributive property of multiplication.
⇒ [tex]270= 598-46x-52x+4x^{2}[/tex]
⇒ [tex]598-98x+4x^{2}= 270[/tex]
Subtracting both side by 270
⇒ [tex]4x^{2}-98x+328= 0[/tex]
using quadratic formula to solve the equation.
⇒ Formula: [tex]\frac{-b\pm \sqrt{b^{2}-4(ac) } }{2a}[/tex]
∴ In the expression , we have a= 4, b= -98 and c= 328.
Now, subtituting the value in the formula.
= [tex]\frac{-(-98)\pm \sqrt{-98^{2}-4(4\times 328) } }{2\times 4}[/tex]
= [tex]\frac{98\pm \sqrt{9604-4(1312) } }{8}[/tex]
Opening parenthesis.
= [tex]\frac{98\pm \sqrt{9604-5248 } }{8}[/tex]
= \frac{98\pm \sqrt{4356}}{8}
We know 66²=4356 and √a²=a or -a
= [tex]\frac{98\pm 66}{8}[/tex]
= [tex]\frac{98+66}{8}\ or \ \frac{98-66}{8}[/tex]
= [tex]20.5 \ or\ 4[/tex]
∴ 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= [tex](23-2x)[/tex]
⇒ Width of rug= [tex]23- 2\times 4[/tex]
⇒ Width of rug= [tex]23-8= 15\ ft[/tex]
Next, Length of rug= [tex](26-2x)[/tex]
⇒ Length of rug= [tex](26-2\times 4)[/tex]
⇒ Length of rug= [tex]26-8= 18\ ft[/tex]
Hence, dimension of rug is [tex](15\times 18)\ ft[/tex]