The depth, hhh metres, of an underwater research drone exploring a lake is modelled by a function h(t)h(t)h(t), where t>0t > 0t>0 is the time in seconds after it passes a specific underwater marker.
The vertical acceleration of the drone is given by the equation
h′′(t)=10t3+12t2 h''(t) = \frac{10}{\sqrt{t^3}} + 12t^2 h′′(t)=t310+12t2A point P(1,5)P(1, 5)P(1,5) lies on the depth-time curve.
Given that the rate of change of depth h′(t)=−2h'(t) = -2h′(t)=−2 at point PPP,
find the equation of the normal to the curve at PPP, writing your answer in the form h=mt+ch = mt + ch=mt+c, where mmm and ccc are constants,
determine an expression for h(t)h(t)h(t).
