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 5.5 days. the average brightness of this star is 5.0 and its brightness changes by ±0.35. using this information, the brightness of the star at time t, where t is measured in days, has been modeled by the function (b(t)=5.0 + 0.35sin(\frac{2pi t}{5.5})). (a) find the rate of change of the brightness at time t days. (\frac{db}{dt}=) (b) find the rate of change of brightness at time (t = 5) days. (\frac{db}{dt}=)

Explanation:

Step1: Differentiate the function

We use the chain - rule. The derivative of a constant is 0, and the derivative of $\sin(u)$ with respect to $t$ is $\cos(u)\cdot\frac{du}{dt}$. Given $B(t)=5.0 + 0.35\sin(\frac{2\pi t}{5.5})$, let $u = \frac{2\pi t}{5.5}$, then $\frac{du}{dt}=\frac{2\pi}{5.5}$. The derivative of the constant 5.0 is 0, and the derivative of $0.35\sin(u)$ is $0.35\cos(u)\cdot\frac{du}{dt}$. So $\frac{dB}{dt}=0 + 0.35\cos(\frac{2\pi t}{5.5})\cdot\frac{2\pi}{5.5}$.
$\frac{dB}{dt}=\frac{0.7\pi}{5.5}\cos(\frac{2\pi t}{5.5})$

Step2: Evaluate at $t = 5$

Substitute $t = 5$ into $\frac{dB}{dt}=\frac{0.7\pi}{5.5}\cos(\frac{2\pi t}{5.5})$.
$\frac{dB}{dt}\big|_{t = 5}=\frac{0.7\pi}{5.5}\cos(\frac{2\pi\times5}{5.5})=\frac{0.7\pi}{5.5}\cos(\frac{20\pi}{11})\approx - 0.39$

Answer:

(a) $\frac{dB}{dt}=\frac{0.7\pi}{5.5}\cos(\frac{2\pi t}{5.5})$
(b) $\frac{dB}{dt}\big|_{t = 5}\approx - 0.39$