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A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by ±0.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function
B(t)=4.0+0.35sin(2πt/5.4)
Find, correct to two decimal places, the rate of increase after one day.

Respuesta :

Answer:

B'(t) = 0.161

Step-by-step explanation:

Data provided in  the question:

The brightness of Delta Cephei at time t, is given by the function:

[tex]B(t)=4.0+0.35\sin(\frac{2\pi t}{5.4})[/tex]

Here, t is in days

Now,.

The rate of change will be B'(t) = [tex]\frac{d(B)}{dt}[/tex]

thus,

[tex]\frac{d(B)}{dt}[/tex] = [tex]0+0.35\cos(\frac{2\pi t}{5.4})\times\frac{d(\frac{2\pi t}{5.4})}{dt}[/tex]

or

[tex]\frac{d(B)}{dt}[/tex] = [tex]0.35\cos(\frac{2\pi t}{5.4})\times\frac{2\pi}{5.4})[/tex]

Now rate of increase after one day i.e after t = 1

we have

[tex]\frac{d(B)}{dt}[/tex] = [tex]0.35\cos(\frac{2\pi(1)}{5.4})\times\frac{2\pi}{5.4})[/tex]

or

[tex]\frac{d(B)}{dt}[/tex] = 0.35 × cos (1.163) × 1.163 [here angle with cos is in radians ]

converting radians to degrees = [tex]1.163\times\frac{\textup{180}^o}{\pi}[/tex]

or

1.163 radians = 66.67°

Therefore,

[tex]\frac{d(B)}{dt}[/tex] = 0.35 × cos (66.67°) × 1.163

= 0.161

The rate of Increase of the brightness of Delta Cephei after one day is; 0.46

We are given that the brightness of the star named Delta Cephei at time t, where t is measured in days, has been modeled by the function;

B(t) = 4.0 + 0.35sin(2πt/5.4)

To find the rate of Increase simply means we have to find the derivative of the function which gives;

B'(t) = (2π/5.4) × 0.35cos (2πt/5.4)

Now, when t = 1 we have;

B'(1) = (2π/5.4) × 0.35cos (2π * 1/5.4)

Now the angle in the bracket is in radians and thus, we will find the cosine of the angle using radians calculator and we have;

B'(1) = (2π/5.4) × 0.3961

B'(1) ≈ 0.46

Read more at; https://brainly.com/question/17110089

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