Calculus Question

Question

During a 10-year study of mosquitoes in a provincial park, the population is best described by the

formula P(t)=20000-4e^(4-7t)+(t^2)e^(7-4t) after t years. Find the minimum population and when it occurred.

Answer 1

For P(t) = 20000 - 4e^4-7t + t²e^7-4t it's pretty easy to see that after just a few years, the function approaches a horizontal asymptote of y = 20,000. Example: consider t = 4. then -4e^4-28 is -1.5 x 10^-10 so VERY close to zero. And 4²e^4-16 is 1.3 x 10^-5 also very close to zero. As "t" increases to 5 then -4e^4-35 is -1.4 x 10^-13 (even closer to zero) and 5²e^4-20 is 2.8 x 10^-6 (also closer to zero) so as t approaches infinity, P(t) approaches 20,000. In fact by 2.5 years, the population IS 20,000 (rounded to the nearest integer).

So we really need to just consider what happens between t = 0 and say t = 2. For this function, taking the derivative and setting it equal to zero to find local extrema is not simple since the resultant derivative is a transcendental function and not solvable as I see it using Algebra. Therefore. since a numeric solution is required anyway, I would suggest you just graph the function and look at it. Doing so, you see there is a local max at t = 0.597 and the lowest function value is at t = 0 where the population is 19,781