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

`(0.7\pi)/(5.3)\cos((2\pi t)/(5.3))`

0.16

Explanation:

According to the question the mathematical model should be

`B(t)=2.9+0.35\sin((2\pi t)/(5.3))`

Differentiating with respect to time we get

`B'(t)=0.35\cos((2\pi t)/(5.3))*((2\pi)/(5.3))\\\Rightarrow B'(t)=(0.7\pi)/(5.3)\cos((2\pi t)/(5.3))`

So, the rate of change of brightness after t days is
`(0.7\pi)/(5.3)\cos((2\pi t)/(5.3))`

After 1 day means
`t=1`

`B'(1)=(0.7\pi)/(5.3)\cos((2\pi* 1)/(5.3))\\\Rightarrow B'(1)=0.16`

The rate of increase after one day is 0.16.