Question

There is a rectangular patch of lawn 20m x 30m. Around the patch of lawn is a rectangular path which in area is the same as one third of the area of the lawn. What is width of the path?

Answer 1

The area of the lawn is \( 600\:m^2\) and the area around is one third of the area \(200\:m^2\). So the two areas put together is the
\[ A_{patch}+A_{lawn} = (2x +20)m*(2x+30)m \]
\[ 800 \: m^2 = (4x^2 + 60x+40x +600)\:m^2\]
\[(4x^2 + 100x +600 -800)\: m^2 = 0 \]
\[(4x^2 + 100x -200)\: m^2 = 0 \]
this is a quadratic equation
\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} = \frac{-100\pm\sqrt{100^2-4*4*(-200)}}{2*4} \]
\[x_1 \approx -26,86\:m\]
\[x_2 \approx 1.86\:m\]
since x cant be negative \(x_2\) must be the solution.