7. In an experiment a scientist considered the loss of mass of a collection of picked leaves. The mass \(M\) grams of a single leaf was measured at times \(t\) days after the leaf was picked. The scientist attempted to find a relationship between \(M\) and \(t\). In a preliminary model she assumed that the rate of loss of mass was proportional to the mass \(M\) grams of the leaf.

1. Write down a differential equation for the rate of change of mass of the leaf, using this model.
2. Show, by differentiation, that \(M = 10 ( 0.98 ) ^ { t }\) satisfies this differential equation. Further studies implied that the mass \(M\) grams of a certain leaf satisfied a modified differential equation $$10 \frac { \mathrm {~d} M } { \mathrm {~d} t } = - k ( 10 M - 1 )$$ where \(k\) is a positive constant and \(t \geq 0\).
   Given that the mass of this leaf at time \(t = 0\) is 10 grams, and that its mass at time \(t = 10\) is 8.5 grams,
3. solve the modified differential equation (I) to find the mass of this leaf at time \(t = 15\).
   7. continued