Question

A Cepheid variable star is a star whose brightness alternately increases and decreases. For a certain star, the interval between times of maximum brightness is 4.2 days. The average brightness of this star is 3.0 and its brightness changes by ±0.25. In view of these data, the brightness of the star at time t, where t is measured in days, has been modeled by the function B(t) = 3.0 + 0.25 sin 2πt 4.2 . (a) Find the rate of change of the brightness after t days. dB dt =

Answer 1

a)
`(dB)/(dt) = (5\pi)/(4.2) \cdot \cos \left(2\pi\cdot (t)/(4.2) \right)`, b)
`(dB)/(dt)\approx 5.595`

Explanation:

a) The rate of change of the brightness of the Cepheid can be determined by deriving the function in time:

`(dB)/(dt) = \left((2\pi)/(4.2) \right)\cdot 0.25\cdot \cos (2\pi\cdot (t)/(4.2))`

`(dB)/(dt) = (5\pi)/(4.2) \cdot \cos \left(2\pi\cdot (t)/(4.2) \right)`

b) The rate of increase after one day is:

`(dB)/(dt) = (5\pi)/(4.2) \cdot \left(2\pi \cdot (1)/(4.2) \right)`

`(dB)/(dt)\approx 5.595`