Science:Math Exam Resources/Courses/MATH102/December 2015/Question 06/Solution 1

This solution will use the theory of Hill functions. An introduction can be found in this video. You may also want to view the second solution presented below.

This is a Hill function with horizontal asymptote $F_{\text{max}}$, half-max $k$, and Hill coefficient 2. For positive values of t, it follows that the function is strictly increasing toward a horizontal asymptote $F_{\text{max}}$.
(a) does not apply because f(t) approaches $F_{\text{max}}$ as $t\to \infty$.
(b) is reasonable. For values of t near zero, f(t) is close to $t^{2}$, which is small near zero. This means that food is gathered slowly at times close to zero.
(c) is not correct: k is the half-max, which means that half of the food will have been taken out of the patch at t = k.
(d) is also not correct: k/2 is even less time than k, so less than half of the food will be recovered in this time.
(e) is wrong; the maximum amount of food available is $F_{\text{max}}$.
answer: $\color {blue}(b)$