2016 NJC CT Q11 [Modified]

The mass, \(x\) grams, of a certain substance present in a chemical reaction at time \(t\) minutes satisfies the differential equation \(\frac{\mathrm{d}x}{\mathrm{d}t}=k\left( 5+8x-4{{x}^{2}} \right)\) , where \(0\le x\le 1\) and \(k\) is a constant. Initially, the mass of the substance is 1 gram and is decreasing at a rate of 0.9 grams per minute at that instant. Show that \(k=-\frac{1}{10}\).[1]

1. By first expressing \(5+8x-4{{x}^{2}}\) in completed square form, find \(t\) in terms of \(x\) exactly. Hence, use your calculator to verify that it takes approximately 1 minute for the substance to become 20% of its initial value.[6]
2. Express the solution of the differential equation in the form \(x=\mathrm{f}(t)\).[2]