a cepheid variable star is a star whose brightness alternately increases and decreases. suppose that cephei…
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 4.1 days. its average brightness is 4 and the brightness changes by +/-0.4. 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 + 0.4sin(2πt/4.1) (a) find the rate of change of the brightness after t days. rate of change = (b) find the rate of increase after one day. rate of increase =
Answer
Explanation:
Step1: Recall the derivative formula
The derivative of $y = a + b\sin(cx)$ with respect to $x$ is $y'=bc\cos(cx)$. Here $a = 4$, $b = 0.4$, and $c=\frac{2\pi}{4.1}$ for $B(t)=4 + 0.4\sin(2\pi t/4.1)$.
Step2: Find the derivative of $B(t)$
Using the formula, $B'(t)=0.4\times\frac{2\pi}{4.1}\cos(2\pi t/4.1)=\frac{0.8\pi}{4.1}\cos(2\pi t/4.1)$. This is the rate - of - change of the brightness after $t$ days.
Step3: Evaluate the derivative at $t = 1$
Substitute $t = 1$ into $B'(t)$. We have $B'(1)=\frac{0.8\pi}{4.1}\cos(2\pi/4.1)$. Calculate $\cos(2\pi/4.1)\approx\cos(1.537)\approx0.087$ and $\frac{0.8\pi}{4.1}\approx0.61$. Then $B'(1)\approx0.61\times0.087 = 0.053$.
Answer:
(a) $\frac{0.8\pi}{4.1}\cos(2\pi t/4.1)$ (b) $0.053$