The radius, \(r\) cm, of a circular chemical stain \(t\) seconds after it starts spreading is modelled by \(r=0.04t^2+0.6t\) and \(0\le t\le15\).
Find \(\dfrac{\mathrm{d}A}{\mathrm{d}t}\) in terms of \(t\), where \(A\) is the area of the stain.[3]
Find the rate of increase of the area of the stain when \(t=10\).[2]
For safety reasons, the laboratory has an automatic alert system. The system is activated if the area of the chemical stain is increasing at a rate greater than 100 cm\(^2\)/s.
This is because the cleaning team needs to respond immediately before the stain spreads beyond the containment mat.
Determine whether the alert system would already have been activated by the time \(t=14\). Justify your answer clearly.[3]