Question

Use a graphing calculator, or another piece of technology, and the following scenario to answer the 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)+8H(x)=6sin⁡(xπ8)+8, where xx is time in hours. Assume x=0x=0 is midnight, x=1x=1 is 11 a.m., x=2x=2 is 22 a.m., etc. When will the waves be at least 88 feet in the next 1212 hours?

Answer 1

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.

Explanation:

The height of the waves on a local beach can be approximated by the function:

`H(x)=6sin(\pi /8\cdot x)+8`

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.

`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`

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.)

`sin(\pi/8\cdot x)=1\\\\\pi/8\cdot x=\pi/2\\\\x=4`

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).