Answer:
Minimum average temperature is 33 °F
Number of cycles of f per month is [tex]\dfrac{1}{12}[/tex]
Time for one full cycle is 12 months
Step-by-step explanation:
[tex]f(t)=59-26 \sin \left(\dfrac{\pi }{6}t\right)[/tex]
where:
- f(t) = the average temperature a month in °F
- t = month of the year
As [tex]-1\leq \sin (x)\leq 1[/tex], the range of the sine part of the function is:
[tex]-26\leq 26\sin \left(\dfrac{\pi }{6}t\right)\leq 26[/tex]
Therefore the minimum temperature will be when
[tex]26\sin \left(\dfrac{\pi }{6}t\right)=26[/tex]
[tex]\implies f(t)=59-26=33 \ \sf \textdegree F[/tex]
The period of the sine curve is the length of one cycle of the curve. The period is the distance between the peak and the next peak (or trough and the next trough).
To find this algebraically, set the sine part of the function to 26 and solve for t:
[tex]\implies 26\sin \left(\dfrac{\pi }{6}t\right)=26[/tex]
[tex]\implies \sin \left(\dfrac{\pi }{6}t\right)=1[/tex]
[tex]\implies \dfrac{\pi }{6}t=\dfrac{\pi }{2} \pm 2\pi n[/tex]
[tex]\implies t=3 \pm 12n[/tex]
Therefore the period of this function is 12 months.
So the number of cycles of f per month is [tex]\dfrac{1}{12}[/tex]
and the time for one full cycle of f is 12 months.
