The length of a rectangle is 10 feet less than 3 times its width. How do you find the dimensions of this rectangle if the area is 48 square feet?

`"let the width "=x`

`"then the length "=3x-10larrcolor(blue)"10 less than 3 times width"`

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

`rArr"area "=x(3x-10)=3x^2-10x`

`"now area "=48`

`rArr3x^2-10x=48larrcolor(blue)"rearrange and equate to zero"`

`3x^2-10x-48=0`

`"the factors of - 144 which sum to - 10 are - 18 and + 8"`

`"splitting the middle term gives"`

`3x^2-18x+8x-48=0larrcolor(blue)"factor by grouping"`

`color(red)(3x)(x-6)color(red)(+8)(x-6)=0`

`(x-6)(color(red)(3x+8))=0`

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

`x-6=0rArrx=6`

`3x+8=0rArrx=-8/3`

`x>0rArrx=6`

`rArr"width "=x=6" feet"`

`rArr"length "=3x-10=18-10=8" feet"`

Let the width `= x` feet

So, length `= 3x-10` feet

Now area of rectangle

`= "length" xx "width"` sq unit.

`= (3x-10)x` sq feet.

Now as per question,

`(3x-10)x = 48`

`rArr 3x^2-10x-48=0`

`rArr 3x^2-(18-8)x-48=0`

`rArr 3x^2-18x+8x-48=0`

`rArr 3x(x-6)+8(x-6)=0`

`rArr (3x+8)(x-6)=0`

`rArr 3x+8=0, x-6=0`

`rArr x = -8/3 and 6.`

Width cannot be negative.

So, `x = 6`

Hence width is `6` feet, and length is `3 xx 6-10 = 8` feet