Question

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?

Answer 1

width = 7 cm
length = 12 cm

Explanation:

It is often helpful to draw a quick sketch.

Let length be `L`
Let width be `w`

Area `=wL`

`= w(2w-2)`

`= 2w^2-2w" "=" "84 cm^2`
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`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)(.)|))`