# 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?

Feb 14, 2018

$\text{length "=8" feet and width "=6" feet}$

#### Explanation:

$\text{let the width } = x$

$\text{then the length "=3x-10larrcolor(blue)"10 less than 3 times width}$

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

$\Rightarrow \text{area } = x \left(3 x - 10\right) = 3 {x}^{2} - 10 x$

$\text{now area } = 48$

$\Rightarrow 3 {x}^{2} - 10 x = 48 \leftarrow \textcolor{b l u e}{\text{rearrange and equate to zero}}$

$3 {x}^{2} - 10 x - 48 = 0$

$\text{the factors of - 144 which sum to - 10 are - 18 and + 8}$

$\text{splitting the middle term gives}$

$3 {x}^{2} - 18 x + 8 x - 48 = 0 \leftarrow \textcolor{b l u e}{\text{factor by grouping}}$

$\textcolor{red}{3 x} \left(x - 6\right) \textcolor{red}{+ 8} \left(x - 6\right) = 0$

$\left(x - 6\right) \left(\textcolor{red}{3 x + 8}\right) = 0$

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

$x - 6 = 0 \Rightarrow x = 6$

$3 x + 8 = 0 \Rightarrow x = - \frac{8}{3}$

$x > 0 \Rightarrow x = 6$

$\Rightarrow \text{width "=x=6" feet}$

$\Rightarrow \text{length "=3x-10=18-10=8" feet}$

Feb 14, 2018

Width$= 6$ feet and length $= 8$ feet

#### Explanation:

Let the width $= x$ feet

So, length $= 3 x - 10$ feet

Now area of rectangle

$= \text{length" xx "width}$ sq unit.

$= \left(3 x - 10\right) x$ sq feet.

Now as per question,

$\left(3 x - 10\right) x = 48$

$\Rightarrow 3 {x}^{2} - 10 x - 48 = 0$

$\Rightarrow 3 {x}^{2} - \left(18 - 8\right) x - 48 = 0$

$\Rightarrow 3 {x}^{2} - 18 x + 8 x - 48 = 0$

$\Rightarrow 3 x \left(x - 6\right) + 8 \left(x - 6\right) = 0$

$\Rightarrow \left(3 x + 8\right) \left(x - 6\right) = 0$

$\Rightarrow 3 x + 8 = 0 , x - 6 = 0$

$\Rightarrow x = - \frac{8}{3} \mathmr{and} 6.$

Width cannot be negative.

So, $x = 6$

Hence width is $6$ feet, and length is $3 \times 6 - 10 = 8$ feet