Chapter 9.4, Problem 73E

To determine

To determine the dimensions of the scarf.

Answer to Problem 73E

Length=60.835inches

Breadth= $6.505$ inches

Explanation of Solution

Given:

Old length=60inches

Old breadth=4inches

Total yarn to knit 396 square inches scarf.

Calculation:

According to question:

Increase in length is 3 times increase in width.

Let the increase in length is $x$ therefore the increase in width is $3x$

Therefore the equation for area of scarf becomes:

  $(60+x)(4+3x)=396$

Opening brackets,

  $240+180x+4x+3x^2=396$

So,

  $240+184x+3x^2=396$

Converting the given equation in standard form i.e. $a{x}^2+bx+c=0$

  $184x+3x^2+240-396=0$

So,

  $3x^2+184x-156=0$

On comparing the equation with $a{x}^2+bx+c$

  $a=3,b=184,c=-156$

Finding the value of $x$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Substituting the values,

  $x=\frac{-(184)\pm \sqrt{{(184)}^2-4(3)(-156)}}{2(3)}$

Opening brackets,

  $x=\frac{-184\pm \sqrt{33856+1872}}{6}$

So,

  $x=\frac{-184\pm \sqrt{35728}}{6}$

Therefore,

  $x=\frac{-184\pm 189.01}{6}$

So,

  $x=\frac{-184+189.01}{6}$ and $x=\frac{-184-189.01}{6}$

So,

  $x=\frac{5.01}{6}$ and $x=-\frac{373.01}{6}$

Increase can never be negative

Therefore,

  $x=0.835$

So,

Length= $x+60$

  $=0.835+60=60.835$ inches

Breadth= $3x+4$

  $=3(0.835)+4=2.505+4$

  $=6.505$ inches