Question

A Cepheid variable star is a star whose brightness alternately increases and decreases. Suppose that Cephei Joe is a star for which the interval between times of maximum brightness is 5.3 days. Its average brightness is 2.9 and the brightness changes by /- 0.35. Using this data, we can construct a mathematical model for the brightness of Cephei Joe at time t, where t is measured in days:______.B(t)=4.2 +0.45sin(2pit/4.4)
(a) Find the rate of change of the brightness after t days.
(b) Find the rate of increase after one day.

Answer 1

Answer:

$\dfrac{0.7\pi}{5.3}\cos(\dfrac{2\pi t}{5.3})$

0.16

Step-by-step explanation:

According to the question the mathematical model should be

$B(t)=2.9+0.35\sin(\dfrac{2\pi t}{5.3})$

Differentiating with respect to time we get

$B'(t)=0.35\cos(\dfrac{2\pi t}{5.3})\times(\dfrac{2\pi}{5.3})\\\Rightarrow B'(t)=\dfrac{0.7\pi}{5.3}\cos(\dfrac{2\pi t}{5.3})$

So, the rate of change of brightness after t days is $\dfrac{0.7\pi}{5.3}\cos(\dfrac{2\pi t}{5.3})$

After 1 day means $t=1$

$B'(1)=\dfrac{0.7\pi}{5.3}\cos(\dfrac{2\pi\times 1}{5.3})\\\Rightarrow B'(1)=0.16$

The rate of increase after one day is 0.16.