Quandaries
and Queries 

Hi, my name is Regis. I am a student. The level of the question is secondary (11). The question is: 

Hi Regis, I really like this problem. I am going to get you started the way I think about this problem and you can continue. If you have problems completing it let me know. (I may have changed your notation slightly.) When I see the instructions to model using a cosine function then, in my mind I see
y = cos(t) The shape of this function is what you need but the axes are in the wrong place and the scales on both axes need to be changed. Your model calls for a repeated pattern starting with the cosine at its minimum (10 degrees C in January), thus the yaxis need to go through the minimum value of the cosine function. You can accomplish this by adding 180^{o} to the xvalues. (From now on I will only plot one cycle of the cosine function.)
y = cos (t + 180^{o}) The scale on the y axis needs adjustment. In the graph above the values of the function are from 1 to 1, a range of 2 units. Your function is to have values from 10 degrees C to 20 degrees C, a range of 30 units. Thus I need to multiply the cosine function by 30/2 = 15. This gives values of the funtion from 15 to 15 and you want 10 to 20, so add 5 to the yvalues. y = 15 cos(t + 180^{o}) + 5 All that remains is the scale on the xaxis. At the moment the one cycle shown on the graph above is for t from 0^{o} to 360^{o}. You need the scale on the xaxis to be in months so that one cycle is from month 0 (January) to month 12 (which is January again). Harley 

