Answer:
so average temperature is 52 °F
Step-by-step explanation:
Given data
function f(t) = 52 + 17 sin πt/12
time = 9 am to 9 pm i.e. ( 0 to 12 ) hours
to find out
average temperature
solution
we know time period is 12 hours so function interval will be ( 0, 12 )
we will integrate the function with respect to t from a to b i.e 0 to 12 limit
i.e.
T_{avg}= [tex]\frac{1}{b-a} \int_{a}^{b} f(t)dt[/tex] .......................1
here put all value and integrate it
T_{avg}= [tex]\frac{1}{12-0} \int_{0}^{12} (52 + 17 sin \pi t/12)dt[/tex]
T_{avg}= [tex]\frac{1}{12} \int_{0}^{12} (52 + 17 sin \pi t/12)dt[/tex]
T_{avg}= [tex]\frac{1}{12} (52t + 17(12/\pi) (-cos \pi t/12)^{12}_0[/tex]
T_{avg}= 0.08333 × (52(12-0) + 204/ [tex]\pi[/tex] (-cos([tex]\pi[/tex] *12)/12) - cos(0))
T_{avg}= 51.99792
so average temperature is 52 °F
