Solar altitude

Question

ℎ()=29.7+23.4⋅cos(2π365(−172))h(t)=29.7+23.4⋅cos⁡(2π365(t−172))

This is a function of the maximum solar altitude in degrees on day t of the year.

How can I find the average maximum height in total for June, July and August using integration, given that June 1st is the 152nd day of the year, and August 31st is the 243rd day of the year?

Comments

The formula doesn’t look right. Shouldn't it be division by 365, not multiplication? And to find the maximum h, we should use differentiation. Integration gives the area under a curve so the units would not make sense in this context. Differentiation would allow the maximum height in degrees to be calculated for the June-August period.

Answer 1

I think you mean h(t)=29.7+23.4cos(2π(x-172)/365). When x=172 the cosine argument=0 and cos(0)=1 so h(172)=29.7+23.4=53.1º.

The day number 172 lies between days 152-243, so the maximum altitude is 53.1º between June and August.