A biologist is monitoring the population of a specialized algae culture, PPP (in thousands), in a controlled environment. At time ttt hours (t>0t > 0t>0), the rate of change of the population is modeled by the derivative:
P′(t)=2t2+at+b3t P'(t) = \frac{2t^2 + at + b}{3\sqrt{t}} P′(t)=3t2t2+at+bwhere aaa and bbb are constants. It is observed that at time t=4t = 4t=4, the population is exactly 10,000 algae (P=10P = 10P=10) and the population is growing at a rate of 5,000 algae per hour (P′=5P' = 5P′=5).
Show that 4a+b=−24a + b = -24a+b=−2.
Given also that a+b=7a + b = 7a+b=7,
Determine, in simplest form, an expression for P(t)P(t)P(t).
A researcher adjusts the experiment such that the new population model is given by G(t)=P(t−3)G(t) = P(t - 3)G(t)=P(t−3). Given that the observation at t=4t = 4t=4 in the original model is recorded as point JJJ,
State the coordinates of the corresponding point KKK on the graph of G(t)G(t)G(t).