Chapter 9.4, Problem 55E

To determine

To determine the width of the border.

Answer to Problem 55E

The width of the border, $x=19.77ft$

Explanation of Solution

Given:

  

Area to cover with patio bricks=140 square feet.

Calculation:

According to question,

Area to cover with patio bricks=140 square feet.

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

For outer square,

Length=20ft

Breadth=16ft

For inner square,

Length=20 $-2x$ ft

Breadth=16 $-2x$ ft

Therefore, area to cover with patio bricks= $(20\times 16)-((20-2x)\times (16-2x))$

On solving brackets,

  $140=(320)-(320-40x-32x+4x^2)$

So,

  $140=320-(320-72x+4x^2)$

Opening brackets,

  $140=-72x+4x^2$

Taking all the terms on one side,

  $4x^2-72x-140=0$

Finding the value of $x$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

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

  $a=4,b=-72,c=-140$

Substituting the values,

  $x=\frac{-(-72)\pm \sqrt{{(-72)}^2-4(4)(-140)}}{2(4)}$

Opening brackets,

  $x=\frac{72\pm \sqrt{5184+2240}}{8}$

So,

  $x=\frac{72\pm \sqrt{7424}}{8}$

Solving root,

  $x=\frac{72\pm 86.16}{8}$

Therefore,

  $x=\frac{72+86.16}{8}$ or $x=\frac{72-86.16}{8}$

So,

  $x=\frac{-14.16}{8}$

and

  $x=\frac{158.16}{8}$

Length can never be negative,

Therefore, $x=19.77ft$