Question

2. (6 points) A 60cm by 80cm photo is to be surrounded with a matting of equal width. The photo and matting is to be covered by a 1m^2 sheet of expensive glass. Find the width of the matting.

Answer 1

Let x be the width of the matting, since it is of equal width and the total area is equal to 10000cm^2 (1m^2=1mx1m=100cmx100cm=10000cm^2); then,

`\begin{gathered} A=(60+2x)(80+2x) \\ \text{and} \\ A=10000 \\ \Rightarrow(60+2x)(80+2x)=10000 \end{gathered}`

In a diagram

Solving for x,

`\begin{gathered} \Rightarrow4x^2+280x+4800=10000 \\ \Rightarrow4x^2+280x-5200=0 \\ \Rightarrow x^2+70x-1300=0 \end{gathered}`

Solving the quadratic equation,

`\Rightarrow x=\frac{-70\pm\sqrt[]{4900+4\cdot1300}}{2}=-35\pm\frac{\sqrt[]{10100}}{2}\Rightarrow-35\pm5\sqrt[]{101}`

And x has to be a positive number since it is a measurement; therefore,

`\begin{gathered} \Rightarrow x=5\sqrt[]{101}-35 \\ \Rightarrow x=15.249378\ldots \\ \Rightarrow x\approx15.25 \end{gathered}`

The answer is approximately 15.25cm.