Chapter 9.4, Problem 56E

To determine

To determine the width of the border.

Answer to Problem 56E

The width of the border, $x=31in$

Explanation of Solution

Given:

  

Area to cover with patio bricks=722 square inches.

Calculation:

According to question,

Area to cover with patio bricks=722 square feet.

Formula for, area to cover with patio bricks= area of outer square- area of inner square

For outer square,

Length=28in

Breadth=22in

For inner square,

Length=28 $-2x$ in

Breadth=22 $-2x$ in

Therefore, area to cover with patio bricks= $(28\times 22)-((28-2x)\times (22-2x))$

On solving brackets,

  $722=(616)-(616-56x-44x+4x^2)$

So,

  $722=616-(616-100x+4x^2)$

Opening brackets,

  $722=-100x+4x^2$

Taking all the terms on one side,

  $4x^2-100x-722=0$

Finding the value of $x$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

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

  $a=4,b=-100,c=-722$

Substituting the values,

  $x=\frac{-(-100)\pm \sqrt{{(-100)}^2-4(4)(-722)}}{2(4)}$

Opening brackets,

  $x=\frac{100\pm \sqrt{10000+11552}}{8}$

So,

  $x=\frac{100\pm \sqrt{21552}}{8}$

Solving root,

  $x=\frac{100\pm 146.80}{8}$

Therefore,

  $x=\frac{100+146.80}{8}$ or $x=\frac{100-146.80}{8}$

So,

  $x=\frac{-46.80}{8}$

and

  $x=\frac{246.80}{8}$

Length can never be negative,

Therefore, $x=30.85in$

Rounding of to nearest $x=31in$