# Question #855d3

Aug 10, 2016

$x = \frac{54 - \sqrt{2196}}{8}$

#### Explanation:

We need to choose and define a variable.

Let the width of the border be $x$ metres.

Now we can define the length and width of the mural, remembering that the border is on both sides.

Length of mural = $\left(15 - 2 x\right)$ metres
width of mural = $\left(12 - 2 x\right)$ metres

The area of the wall is $15 \times 12 = 180 {m}^{2}$

The area of the mural is 75% of this: $0.75 \times 180 = 135 {m}^{2}$

Now we have enough information to write an equation and solve it.

$A r e a = l \times b = 135$

$\left(15 - 2 x\right) \left(12 - 2 x\right) = 135$

$180 - 30 x - 24 x + 4 {x}^{2} - 135 = 0$

$4 {x}^{2} - 54 x + 45 = 0$

Using the formula:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$x = \frac{- \left(- 54\right) \pm \sqrt{{\left(- 54\right)}^{2} - 4 \left(4\right) \left(45\right)}}{2 \times 4}$

$x = \frac{54 \pm \sqrt{{\left(- 54\right)}^{2} - 720}}{8}$

$x = \frac{54 \pm \sqrt{2196}}{8}$
Estimating which value we should use gives the following:

$x \approx \frac{54 + 50}{8} = \approx 13$

$x \approx \frac{54 - 50}{8} \approx 0.5$

Obviously we cannot subtract 13 twice from either the length or the width, so the second answer is the one we need.