7. Two ramblers, Alison and Bill, are out walking. At midday, Alison is at the fixed origin \(O\), and Bill is at the point with position vector \(\left( { } ^ { - } 5 \mathbf { i } + 12 \mathbf { j } \right) \mathrm { km }\) relative to \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular, horizontal unit vectors. They are both walking with constant velocity - Alison at \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\), and Bill at a speed of \(2 \sqrt { } 10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a direction parallel to the vector ( \(3 \mathbf { i } + \mathbf { j }\) ).

1. Find the distance between the two ramblers at midday.
2. Show that the velocity of Bill is \(( 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
3. Show that, at time \(t\) hours after midday, the position vector of Bill relative to Alison is $$[ ( 4 t - 5 ) \mathbf { i } + ( 12 - 3 t ) \mathbf { j } ] \mathrm { km } .$$
4. Show that the distance, \(d \mathrm {~km}\), between the two ramblers is given by $$d ^ { 2 } = 25 t ^ { 2 } - 112 t + 169$$
5. Using your answer to part (d), find the length of time to the nearest minute for which the distance between the Alison and Bill is less than 11 km .