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

Mar 13, 2018

$l = 27 c m$
$w = 12 c m$

#### Explanation:

$l = \text{length}$
$w = \text{width}$

$l = 3 \cdot w - 9$
$A = \textcolor{red}{l} \cdot w$
$A = \textcolor{red}{\left(3 \cdot w - 9\right)} \cdot w$
324=3w^2-9w|:3
$108 = {w}^{2} - 3 w | - 108$
$0 = {w}^{2} - 3 w - 108$
$0 = {\left(w - \frac{3}{2}\right)}^{2} - 108 - \frac{9}{4} | + \frac{441}{4}$
$\frac{441}{4} = {\left(w - \frac{3}{2}\right)}^{2} | \sqrt{}$
$\pm \frac{21}{2} = w - \frac{3}{2} | + \frac{3}{2}$
$\frac{3}{2} \pm \frac{21}{2} = w$
${w}_{1} = 12 c m \mathmr{and} \cancel{{w}_{2} = - 9 c m}$

$l = 3 \cdot 12 - 9 = 27 c m$

Mar 13, 2018

$\text{length "=27" cm and width "=12" cm}$

#### Explanation:

$\text{let the width } = w$

$\text{then length "l=3w-9to" 9 less than 3 times width}$

$\text{area of rectangle "="length"xx"width}$

$\Rightarrow \text{area } = w \left(3 w - 9\right)$

${\text{now area "=324" cm}}^{2}$

$\Rightarrow w \left(3 w - 9\right) = 324$

$\text{distribute and equate to zero}$

$\Rightarrow 3 {w}^{2} - 9 w - 324 = 0 \leftarrow \textcolor{b l u e}{\text{in standard form}}$

$\Rightarrow 3 \left({w}^{2} - 3 w - 108\right) = 0$

$\text{factor "w^2-3w-108" using the a-c method}$

$\text{The factors of - 108 which sum to - 3 are - 12 and + 9}$

$\Rightarrow 3 \left(w - 12\right) \left(w + 9\right) = 0$

$\text{equate each factor to zero and solve for w}$

$w - 12 = 0 \Rightarrow w = 12$

$w + 9 = 0 \Rightarrow w = - 9$

$\text{but } w > 0 \Rightarrow w = 12$

$\text{Hence width "=w=12" cm and}$

$\text{length "=3w-9=(3xx12)-9=27" cm}$