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An advertising screen is going to be installed in a shopping centre. The LCD part of screen is $6.5 m$ wide and $2.5 m$ tall, and is surrounded by a plastic chassis which is $x m$ wide on three edges of the screen, and $2 x m$ wide on the bottom edge.

Advertising screen

It is known that the total area of the screen (the LCD and the chassis) is $20 m^{2}$ .

Find the value of $x$ in meters to $3$ significant figures.

Hint

The total height of the screen is $3 x + 2.5$ and the total width is $2 x + 6.5$

Solution

The total height of the screen is $3 x + 2.5$ and the total width is $2 x + 6.5$ , so the total area is

\begin{aligned} (3 x + 2.5) (2 x + 6.5) & = 20 \\ 6 x^{2} + 24.5 x + 16.25 & = 20 \\ 6 x^{2} + 24.5 x - 3.75 & = 0 \\ 24 x^{2} + 98 x - 15 & = 0 \end{aligned}

so by the quadratic formula

\begin{aligned} x & = \frac{- 98 \pm \sqrt{9604 - 4 (24) (- 15)}}{48} \\ = \frac{- 49 \pm \sqrt{2761}}{24} \end{aligned}

We discard the negative solution, and find $x$ to be $0.148 m$