Answer:
The waves will be over 12 feet between 1:51 am (x=1.86) and 6:08 am (x=6.14) in the first 12 hours of the day.
Step-by-step explanation:
The height of the waves on a local beach can be approximated by the function:
[tex]H(x)=6sin(\pi /8\cdot x)+8[/tex]
H is given in feet and t is given in hours of the day.
We have to calculate when will the waves be at least 8 feet in the next 12 hours.
[tex]H(x)=6sin(\pi /8\cdot x)+8\geq12\\\\6sin(\pi /8\cdot x)\geq 12-8\\\\sin(\pi /8\cdot x)\geq4/6\\\\sin(\pi /8\cdot x)\geq2/3\\\\\pi /8\cdot x\geq arcsin(2/3)\\\\\pi /8\cdot x\geq 0.73\\\\x\geq 1.86[/tex]
At x=1.86 (1:51 a.m.) the waves are 12 ft high and going up.
The maximum height happens at x=4 (4 a.m.)
[tex]sin(\pi/8\cdot x)=1\\\\\pi/8\cdot x=\pi/2\\\\x=4[/tex]
From x=1.86 to x=4 we have 2.14 hours. Then, as this is a symetric function with axis x=4, at x=(4+2.14)=6.14 hours (6:08 a.m.), the waves will be 12 feet high again, and then going below 12 feet after that.
Then, the waves will be over 12 feet between 1:51 am (x=1.86) and 6:08 am (x=6.14).
