8. Some ink is poured onto a piece of cloth forming a stain that then spreads.
The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by
$$A = ( p + q t ) ^ { 2 } ,$$
where \(p\) and \(q\) are positive constants.
Given that when \(t = 0 , A = 4\) and that when \(t = 5 , A = 9\),
- find the value of \(p\) and show that \(q = \frac { 1 } { 5 }\),
- find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
- find the rate at which the area of the stain is increasing when \(t = 15\).