Answer: After about 9.03 hours the temperature first reach 82 degrees.
Step-by-step explanation:
The sinusoidal function is given by :
[tex]y=A\sin[\omega(x-\alpha)]+C[/tex]
where, A = amplitude; [tex]\omega=\dfrac{2\pi}{period}[/tex] , α= phase shift on the Y-axis and C = midline.
As per given,
Average daily temperature= [tex]C=\dfrac{73+97}{2}=85[/tex] [midline is average of upper and lower limit.]
A= 97-85 = 12
Phase shift: [tex]\alpha=10[/tex]
Period = 24 hours;
[tex]\omega=\dfrac{2\pi}{24}=\dfrac{\pi}{12}[/tex]
Substitute all values in sinusoidal function, we get
[tex]y=12\sin[\dfrac{\pi}{12}(x-10)]+85[/tex]
Put y= 82, we get
[tex]82=12\sin[\dfrac{\pi}{12}(x-10)]+85\\\\\Rightarrow\ -3= 12\sin[\dfrac{\pi}{12}(x-10)]\\\\=\dfrac{-1}{4}= \sin[\dfrac{\pi}{12}(x-10)]\\\\\Rightarrow\ \dfrac{\pi}{12}(x-10)=\sin^{-1}(\dfrac{-1}{4})\\\\\Rightarrow\ x-10=\dfrac{12}{\pi}(\sin^{-1}(\dfrac{-1}{4}))\\\\\Rightarrow\ x=\dfrac{12}{\pi}(\sin^{-1}(\dfrac{-1}{4}))+10\\\Rightarrow\ x\approx9.03[/tex]
Hence, After about 9.03 hours the temperature first reach 82 degrees.
