Precalc question?

Question

Nadia is a robot that can continuously adjust the length of its hair so that its length is a sinusoidal function of time. At its shortest, it's 6 cm long. At its longest, it's 24 cm long. Today, Nadia's hair is 15 cm long and the length is decreasing. The hair will decrease in shortness until it reaches the minimum length exactly four days from now.

a) Find the amplitude, period, phase shift, and mean

b) When will Nadia's hair be 21cm long (PRINCIPAL SOLUTION)? Give your answer to the nearest three decimal places.

c) When will Nadia's hair be 21cm long (SYMMETRY SOLUTION)? Give your answer to the nearest three decimal places. Attempt to limit any rounding error by hanging on to 6+ decimals throughout the problem.

Answer 1

So you'll be making a wave where the x-axis is time and the y-axis is hair length. I've made a list of how to find each piece of information in part a:

1. The amplitude of a wave is calculated using the equation A = (highest point of the wave - lowest point of the wave) / 2.
2. The mean (which I assume means the midline) is the same formula as amplitude but you change the "-" sign to a "+" sign.
3. The period is how far the wave extends until it repeats itself. It can also be thought of as the distance between two crests (high points) of the wave. In the problem, they tell you that it is 4 days between the hair being 15 cm and 6 cm. If you can figure out which fraction of the wave that distance covers (e.g. 1/2 of the wave, 1/3 wave, or something else) then you can figure out the period of the wave.
4. The phase shift depends on where your wave starts. It sounds like the problem wants you to start from the 15 cm hair length since it says "today". This means you want the wave to be at 15 cm going down when it crosses the y-axis. I think the easiest way to do this is to write an equation for your wave (using the amplitude, mean, and period you found earlier), graph it, then see how far left or right you need to shift it for the 15 cm portion to line up with the y-axis.

The period and phase shift might be a little bit harder to understand without a visual aid so let me know if you get stuck somewhere or need help writing an equation!

Once you have done part a, you can find the answer to part b by figuring out where your graph/equation hits 21 cm. The term "principal solution" means the answers within the first period of the equation, i.e. how many times the wave hits 21 cm before the wave starts to repeat itself.

I'm not really sure what your teacher means by symmetric solution, so I don't think I can help you there without clarification.