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
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