Answer
(a) The rate of change of brightness after $t$ days is $$\frac{0.7\pi}{5.4}\cos(\frac{2\pi t}{5.4})$$
(b) The rate of increase after 1 day is $0.16$.
Work Step by Step
$$B(t)=4.0+0.35\sin(\frac{2\pi t}{5.4})$$
(a) The rate of change of brightness after $t$ days is in fact the derivative of the formula of the star brightness given.
Therefore, to find the rate of change, we need to find the derivative of $B(t)$, or in other words, $B'(t)$. $$B'(t)=[4.0+0.35\sin(\frac{2\pi t}{5.4})]'$$ $$B'(t)=0+0.35\frac{d\sin(\frac{2\pi t}{5.4})}{dt}$$ $$B'(t)=0.35\frac{d\sin(\frac{2\pi t}{5.4})}{d(\frac{2\pi t}{5.4})}\frac{\frac{2\pi}{5.4}dt}{dt}$$ $$B'(t)=0.35\cos(\frac{2\pi t}{5.4})\frac{2\pi}{5.4}$$ $$B'(t)=\frac{0.7\pi}{5.4}\cos(\frac{2\pi t}{5.4})$$
Therefore, the rate of change of brightness after $t$ days is $$\frac{0.7\pi}{5.4}\cos(\frac{2\pi t}{5.4})$$
(b) The rate of increase after 1 day, which means $t=1$, is $$\frac{0.7\pi}{5.4}\cos(\frac{2\pi\times1}{5.4})$$ $$=0.407\cos(1.164)$$ $$=0.407\times0.396$$ $$\approx0.16$$
