Question

A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta, Cephei, the time between period of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by \( \pm 0.35\) magnitude. Find a function that models the brightness of Delta Cephei as a function of time.

Short answer

The function that models the brightness of Delta Cephei as a function of time is \(M\left( t \right) = 4.0 + 0.35\sin \left( {\frac{{2\pi }}{{5.4}}t} \right)\).

Step-by-step solution

Observe the given data

It is convenient to model the brightness of Delta Cephei as a function of time with the use of the sine function.

Consider the time period to be 5.4 days. On the scale of magnitude, let amplitude to be 0.35 and average magnitude to be 4.0.

Determine the function that model the brightness of Delta Cephei

Let \(t = 0\) be the time of average brightness, then the magnitude of brightness can be written as a function of time \(t\) in days.

It can be modeled by the formula as shown below:

\(M\left( t \right) = 4.0 + 0.35\sin \left( {\frac{{2\pi }}{{5.4}}t} \right)\)

Thus, the function that models the brightness of Delta Cephei as a function of time is \(M\left( t \right) = 4.0 + 0.35\sin \left( {\frac{{2\pi }}{{5.4}}t} \right)\).