An ecologist monitors the population of Arctic hares, HHH hundreds, on a remote island at various times ttt months after a monitoring project began. The recorded data follows a cyclic pattern based on seasonal availability of vegetation. The researcher models this population using the equation:
H(t)=6.52cos(2π(t−2.5)10.5)+14.34 H(t) = 6.52 \cos\left( \frac{2\pi(t - 2.5)}{10.5} \right) + 14.34 H(t)=6.52cos(10.52π(t−2.5))+14.34Find the minimum population of hares predicted by the model. Give your answer to the nearest whole hare.
Find the duration of the longest continuous period during the first 21 months of the project (from t=0t=0t=0 to t=21t=21t=21) where the population predicted by the model is above 18 hundred. Give your answer in months to one decimal place.
A different colony of hares in a more sheltered valley is found to have a more stable population with less variation between peaks and troughs, though it maintains the same average population and peak times. A student suggests refining the model for this sheltered colony by increasing the value 6.52 to 7.80. Explain whether this refinement is appropriate.
