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

Answer:

a) $\frac{dB}{dt} = \frac{5\pi}{4.2} \cdot \cos \left(2\pi\cdot \frac{t}{4.2} \right)$, b) $\frac{dB}{dt}\approx 5.595$

Step-by-step explanation:

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

$\frac{dB}{dt} = \left(\frac{2\pi}{4.2} \right)\cdot 0.25\cdot \cos (2\pi\cdot \frac{t}{4.2})$

$\frac{dB}{dt} = \frac{5\pi}{4.2} \cdot \cos \left(2\pi\cdot \frac{t}{4.2} \right)$

b) The rate of increase after one day is:

$\frac{dB}{dt} = \frac{5\pi}{4.2} \cdot \left(2\pi \cdot \frac{1}{4.2} \right)$

$\frac{dB}{dt}\approx 5.595$