In question 31 what is the minimum snowfall and in which years does it occur?

Answer 1

The average annual snowfall in a certain region is modeled by the function below, where S represents the annual snowfall, in inches, and t represents the number of years since 1970. What is the minimum annual snowfall, in inches, for this region?S(t) = 20 + 10cos(t x pi/5) The first thing to realise is that the value of the cosine function can never be less than -1, regardless of what it is being applied to. Therefore the minimum value of the function is S(t) = 20 + (10 x [-1]) = 10 inches. ---- In which years between 1970 and 2000 did the minimum amount of snow fall? We are now required to deduce in which years this snowfall actually occured. We know that they are years such that cos(t x pi/5) = -1 Taking the inverse cosine of both sides: t x pi/5 = pi therefore t1 = 5 and the first year with minimum snowfall is 1975. However, we need to remember that trigonometric equations such as the one we have just solved have an infinite number of solutions. If our function had merely involved cos(t) then its period would have been 2pi, and the minimum amount of snow would have fallen every 2pi years. However, we actually have cos(t x pi/5) which fluctuates pi/5 times as fast as cos(t). It follows that the period of our function is 2pi/(pi/5) which equals 10. So minimum snowfall will occur in 1975 and again in 1985 and 1995.