The length of a rectangle is 2 centimeters less than twice the width. If the area is 84 square centimeters how do you find the dimensions of the rectangle?

1 Answer
Nov 14, 2016

width = 7 cm
length = 12 cm

Explanation:

It is often helpful to draw a quick sketch.

Let length be #L#
Let width be #w#

Tony B

Area #=wL #

#= w(2w-2) #

#= 2w^2-2w" "=" "84 cm^2#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine "w)#

Subtract 84 from both sides

#0=2w^2-2w-84" " larr" this is a quadratic"#

I take one look at this and think: 'can not spot how to factorise so use the formula.'

Compare to #y=ax^2+bx+c" "# where #" "x=(-b+-sqrt(b^2-4ac))/(2a)#

So for our equation we have:

#a=2"; "b=-2"; "c=-84#

#=>w=(2+-sqrt(2^2-4(2)(-84)))/(2(2))#

#w=(2+-sqrt(676))/4#

#w=2/4+-26/4#

To have #w# as a negative value is not logical so go for:

#" "color(green)(ul(bar(|color(white)(.)w=1/2+6 1/2=7 cmcolor(white)(.)|))#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine "L)#

#L=2w-2# so substitute for #w# giving:

#L=2(7)-2 = 12 cm#

#" "color(green)(ul(bar(|color(white)(./.)L=2(7)-2 = 12 cmcolor(white)(.)|))#