Question

Siree wants to go surfing for one hour tomorrow. The height of the waves on a local beach can be approximated by the function H(x)=6sin(xπ8)+8, where x is time in hours.
Assume x=0 is midnight, x=1 is 1 a.m., x=2 is 2 a.m., etc.


When will the waves be at least 8 feet in the next 12 hours?


A.The height of the waves are at least 8 feet between at 4:00 a.m. and 12:00 p.m.


B. The height of the waves are at least 8 feet between at 12:00 a.m. and 8:00 a.m.


C.The height of the waves are at least 8 feet between at 4:00 p.m. and 8:00 p.m.


D.The height of the waves are at least 8 feet between at 8:00 a.m. and 4:00 p.m.

Please help, I don't know where to start! Can I get the brainliest person?

Answer 1

Answer:

• Option B. The height of the waves are at least 8 feet between 12:00 am and 8:00 am.

Explanation:

The correct function H(x) that approximate the height of the waves is:

• $H(x)=6sin(\frac{x\pi}{8})+8$

Now, understand what each term and factor mean:

1. Coefficient 6:

• 6 is the amplitude of the wave. It means that the crest of the wave is 6 feet above the rest point, and the trough is 6 feet below the rest point.

2. Constant term 8:

• 8 is the height of the rest point of the wave. That means that the crest of the wave will b 6 feet + 8 feet = 14 feet; and the trough will be - 6 feet + 8 feet = 2 feet.

        This is the wave will be oscilating between 2 and 14 feet.

3. Sine part of the function:

• $sin(\frac{x\pi }{8})$

The period of the sine function is given by:

• $\frac{2\pi }{\pi/8 }=16$

That means that the pattern of the wave will repeat every 16 hours (period = 16 hours).

4. Calculate the height of the wave at midnight, i.e x = 0

• $H(0)=6sin(0)+8=8$

Then, then the waves will be 8 feet the first time at midnight.

5. Find the time when the wave reaches the maximumm height first time:

• H(max) = 6 feet + 8 feet = 14

• $14=6sin(\frac{x\pi }{8})+8\\\\ 6=6sin(\frac{x\pi }{8})\\\\sin(\frac{x\pi }{8})=1\\\\ \frac{x\pi }{8}=\frac{\pi }{2}\\\\ x=4$

Hence, the wave will be at its maximum height (14 feet) at 4 am.

6. Use symmetry.

The wave tool 4 hours to go from 8 feet to 16 feet; given the symmetry of the function it will take other 4 hours to fall from 16 feet to 8 feet.

Hence, the wave will be 8 feet again at 4am + 4 = 8 am.

7 Conclusion:

• The height of the waves will be at least 8 feet between 12:00 am (midnight) and 8:00 am. This is the option B.