Question

A rectangle contains 324 sq cm. If the length is nine less than three times the width, what are the dimensions of the rectangle?

Answer 1

`l=27cm`
`w=12cm`

Explanation:

`l="length"`
`w="width"`

`l=3*w-9`
`A=color(red)(l)*w`
`A=color(red)((3*w-9))*w`
`324=3w^2-9w|:3`
`108=w^2-3w|-108`
`0=w^2-3w-108`
`0=(w-3/2)^2-108-9/4|+441/4`
`441/4=(w-3/2)^2|sqrt()`
`+-21/2=w-3/2|+3/2`
`3/2+-21/2=w`
`w_1=12cm or cancel(w_2=-9cm)`

`l=3*12-9=27cm`

Answer 2

`"length "=27" cm and width "=12" cm"`

Explanation:

`"let the width "=w`

`"then length "l=3w-9to" 9 less than 3 times width"`

`"area of rectangle "="length"xx"width"`

`rArr"area "=w(3w-9)`

`"now area "=324" cm"^2`

`rArrw(3w-9)=324`

`"distribute and equate to zero"`

`rArr3w^2-9w-324=0larrcolor(blue)"in standard form"`

`rArr3(w^2-3w-108)=0`

`"factor "w^2-3w-108" using the a-c method"`

`"The factors of - 108 which sum to - 3 are - 12 and + 9"`

`rArr3(w-12)(w+9)=0`

`"equate each factor to zero and solve for w"`

`w-12=0rArrw=12`

`w+9=0rArrw=-9`

`"but "w>0rArrw=12`

`"Hence width "=w=12" cm and"`

`"length "=3w-9=(3xx12)-9=27" cm"`