6 A cyclist and his machine have a total mass of 80 kg . The cyclist starts from rest and rides from the bottom to the top of a straight slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). The cyclist exerts a constant force of magnitude 120 N . There is a resisting force of magnitude \(8 v \mathrm {~N}\) acting on the cyclist, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after the start.

1. Show that \(\left( \frac { 1 } { 5 - v } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 10 }\).
2. Solve this differential equation and hence show that \(v = 5 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 10 } t } \right)\).
3. Given that the cyclist takes 20 s to reach the top of the slope, find the length of the slope.