The temperature (Fahrenheit) in a certain city t hours after 9 a.m. was approxim

Question

The temperature (Fahrenheit) in a certain city t hours after 9 a.m. was approximated by T=50 +14sin ( ?t/12) Find the average temperature during the period from 9 a.m. to 9 p.m.

Explanation / Answer

a) The two extrema are t = 6 and 18 hour, since the interval is from 9 am to 9 pm with half the period, 6 is the max. Check the end points t = 0 and 12, Both give you 50 degrees as min. b) Taver= 1/12 *(integral 0->12)T(t)dt (average value theorum),will give you 58.9 degrees (This is the definition the weather bureau uses for "average temperature") (Tmax + Tmin )/2 = (64+50)/2 NOT = 58.9 degrees c) If T(t) is a linear function of t T(t) = mt + b T(9am) = b (y-intercept),which is the Tmin. T(3pm) = m(t) + b, which is Tmax Average T = Area of the trapezoid /(3pm-9am) Area of the trapezoid = (1/2)(6hour)(m(t) + b + b) Average T = (1/2)(6hour)(m(t) + b + b) /(6hours) Average T=(1/2)(m(t) + b + b ) = (1/2)(Tmax + Tmin)

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