Answer
the brightness of Delta Cephei at time $t$ has been modeled by the function:
$$
B(t)=4.0+0.35 \sin \frac{2 \pi t}{5.4}
$$
where $t $ is measured in days.
(a) the rate of change of the brightness after t days is given by:
$$
\begin{aligned}
\frac{d B}{d t} &=\left(0.35 \cos \frac{2 \pi t}{5.4}\right)\left(\frac{2 \pi}{5.4}\right)\\
&=\frac{0.7 \pi}{5.4} \cos \frac{2 \pi t}{5.4}\\
&=\frac{7 \pi}{54} \cos \frac{2 \pi t}{5.4}
\end{aligned}
$$
(b) the rate of increase after one day.
At $ t=1,$
$$ \frac{d B}{d t}=\frac{7 \pi}{54} \cos \frac{2 \pi}{5.4} \approx 0.16$$
Work Step by Step
the brightness of Delta Cephei at time $t$ has been modeled by the function:
$$
B(t)=4.0+0.35 \sin \frac{2 \pi t}{5.4}
$$
where $t $ is measured in days.
(a) the rate of change of the brightness after t days is given by:
$$
\begin{aligned}
\frac{d B}{d t} &=\left(0.35 \cos \frac{2 \pi t}{5.4}\right)\left(\frac{2 \pi}{5.4}\right)\\
&=\frac{0.7 \pi}{5.4} \cos \frac{2 \pi t}{5.4}\\
&=\frac{7 \pi}{54} \cos \frac{2 \pi t}{5.4}
\end{aligned}
$$
(b) the rate of increase after one day.
At $ t=1,$
$$ \frac{d B}{d t}=\frac{7 \pi}{54} \cos \frac{2 \pi}{5.4} \approx 0.16$$
