The velocity, vvv m s−1\text{m s}^{-1}m s−1, of a research drone at time ttt seconds (t>0t > 0t>0) is defined such that its acceleration aaa follows the equation:
a=dvdt=12t+kt3+10 a = \frac{dv}{dt} = \frac{12}{\sqrt{t}} + \frac{k}{t^3} + 10 a=dtdv=t12+t3k+10where kkk is a constant.
It is observed that the rate of change of the drone's acceleration is zero at the instant where t=4t = 4t=4.
Determine the value of kkk.
Given that the drone's velocity is 100100100 m s−1\text{m s}^{-1}m s−1 when t=1t = 1t=1, find an expression for vvv in terms of ttt, giving each term in simplest form.
