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 6 days. Its average brightness is 3.5 and the brightness changes by /-0.25. 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:

a) The rate of change of the brightness after t days is $B^{\prime}(t) = 0.204525\pi\cos{(0.4545\pi t)}$

b) The rate of increase after one day is of 0.0915.

Step-by-step explanation:

The brightness after t days is given by:

$B(t) = 4.2 + 0.45\sin{(\frac{2\pi t}{4.4})} = 4.2 + 0.45\sin{(0.4545\pi t)}$

A) Find the rate of change of the brightness after t days.

This is $B^{\prime}(t)$

The derivative of a constant is 0, the derivative of $\sin{at}$ is $a\cos{at}$

So, in this case, we have that:

$B^{\prime}(t) = 0.45*0.4545\pi\cos{(0.4545\pi t)} = 0.204525\pi\cos{(0.4545\pi t)}$

The rate of change of the brightness after t days is $B^{\prime}(t) = 0.204525\pi\cos{(0.4545\pi t)}$

B) Find the rate of increase after one day.

This is $B^{\prime}(1)$. So

$B^{\prime}(1) = 0.204525\pi\cos{(0.4545\pi)} = 0.0915$

The rate of increase after one day is of 0.0915.