Question

Astronomers have noticed that the number of visible sunspots varies from a minimum of about 20 to a maximum of about 130 per year, with a sinusoidal pattern that has a repeating period every 11 years. Assume we started tracking the data in 2000 when the number of sunspots was at a maximum. Write a cosine function n(t) which models this phenomenon in terms of the time t which represents the years since 2000.

Answer 1

The cosine function that models this situation is:

$f(n) = 55\cos{\frac{2\pi}{11}n} + 75$

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A standard cosine equation is modeled by:

$y = A\cos{Bx} + C$

In which:

• A is half the amplitude, which is between -1 and 1.
• The period is $\frac{2\pi}{B}$
• C is the vertical shift.

Minimum of about 20 to a maximum of about 130 per year:

Thus the amplitude is:

$A = \frac{130 - 20}{2} = 55$

This would give a function between -55 and 55, and we want between 20 and 130, thus as vertical shift of 75, which means that $C = 75$.

Repeating period every 11 years.

Thus:

$\frac{2\pi}{B} = 11$

$11B = 2\pi$

$B = \frac{2\pi}{11}$

Then

$f(n) = 55\cos{\frac{2\pi}{11}n} + 75$

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