Sunspot Problem: For several hundred years, astronomers have kept track of the number of solar flares, or "sunspots," which occur on the surface of the sun. The number of sunspots counted in a given year varies from a minimum of about 10 per year to a maximum of 110 per year. Between the maximums that occurs in the years 1750 and 1948, there were 18 complete cycles.
a) What is the period of the sunspot cycle?
b) Assume that the number of sunspots varies sinusoidally with the year. *FIND THE EQUATION IF 1948 HAS A MAXIMUM*
c) HOW MANY SUNSPOTS IN 2020? THIS YEAR?
d) WHAT IS THE FIRST YEAR AFTER 2020 THAT 35 SUNSPOTS WILL OCCUR?
e) WHAT IS THE FIRST YEAR AFTER 2020 THAT THERE WILL BE A MAXIMUM?
a) What is the period of the sunspot cycle?
b) Assume that the number of sunspots varies sinusoidally with the year. *FIND THE EQUATION IF 1948 HAS A MAXIMUM*
c) HOW MANY SUNSPOTS IN 2020? THIS YEAR?
d) WHAT IS THE FIRST YEAR AFTER 2020 THAT 35 SUNSPOTS WILL OCCUR?
e) WHAT IS THE FIRST YEAR AFTER 2020 THAT THERE WILL BE A MAXIMUM?
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Hello
There is a maximum at 1948, and between 1750 and 1948 there were 18 complete cycles, so
there was a maximum also in 1750.
The sin curve has to be extended, so that one cycle = (1948-1750)/18 = 11 years
a) Period time = 11 years)
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The curve has to be shifted to the left by a quarter period time to have a maximum in 1750.
The amplitude has to be enlarged from 1 to (110-10)/2 = 50 years, the axis has to be lifted by
60 years to have maxima and minima at 110 and 10.
This yields the equation of the sunspots:
b) Equation: y = 50sin(2pi/11(x+11/4)) + 60
--------------------------------------…
2020 is 270 years after 1750, so plug in 270 for x and find y:
c) number of sunspots in2020 (x = 270) = 12,0254
for this year: plug in 261 for x
--------------------------------------…
2020 is 270/11 periods after 1750 = 24,545 periods.
e) The next maximum is at 25 periods after 1750 = 25*11 years after 1750
= 1750 + 275 = in 2025
--------------------------------------…
d) plug in 35 for y : the corresponding x = 11/3 years +- n*11 years. The next maximum after 2020
was in 2025. And the next time after 2020 when there are 35 sunspots is therefore
11/3 years ( = 3 2/3 years) before the next maximum, i.e., before 2025. And this is in the year
2021
--------------------------------------…
Regards
There is a maximum at 1948, and between 1750 and 1948 there were 18 complete cycles, so
there was a maximum also in 1750.
The sin curve has to be extended, so that one cycle = (1948-1750)/18 = 11 years
a) Period time = 11 years)
-------------------------------------
The curve has to be shifted to the left by a quarter period time to have a maximum in 1750.
The amplitude has to be enlarged from 1 to (110-10)/2 = 50 years, the axis has to be lifted by
60 years to have maxima and minima at 110 and 10.
This yields the equation of the sunspots:
b) Equation: y = 50sin(2pi/11(x+11/4)) + 60
--------------------------------------…
2020 is 270 years after 1750, so plug in 270 for x and find y:
c) number of sunspots in2020 (x = 270) = 12,0254
for this year: plug in 261 for x
--------------------------------------…
2020 is 270/11 periods after 1750 = 24,545 periods.
e) The next maximum is at 25 periods after 1750 = 25*11 years after 1750
= 1750 + 275 = in 2025
--------------------------------------…
d) plug in 35 for y : the corresponding x = 11/3 years +- n*11 years. The next maximum after 2020
was in 2025. And the next time after 2020 when there are 35 sunspots is therefore
11/3 years ( = 3 2/3 years) before the next maximum, i.e., before 2025. And this is in the year
2021
--------------------------------------…
Regards