An autonomous underwater vehicle (AUV) is descending into a lake. Its depth D D\,D meters relative to its starting point at time t t\,t seconds is modelled by the function D(t)D(t)D(t) for t>0t > 0t>0.
The rate of change of the AUV's depth is given by
D′(t)=3t+4−16t2 D'(t) = 3\sqrt{t} + 4 - \frac{16}{t^2} D′(t)=3t+4−t216At t=4t = 4t=4, the AUV is at a depth of 15 meters, represented by the point P(4,15)P(4, 15)P(4,15) on the curve y=D(t)y = D(t)y=D(t).
Determine the equation of the normal to the curve y=D(t)y = D(t)y=D(t) at the point PPP. Give your answer in the form at+by+c=0at + by + c = 0at+by+c=0, where a,b, a, b,\,a,b, and c c\,c are integers.
Find the expression for D(t)D(t)D(t).
