Scientists believe that the average temperature at various places on Earth vary from cooler to warmer over thousands of years of gradual climate change. Suppose that at one place, the highest avg temp is 80 and the lowest is 60. Also suppose that the time it takes to go from the high to the low average is 20,000 years and in the year 2000 the avg temp is at a high point of 80. How can we use a sinusoidal expression to model this phenomenon?
My work:
So far I have
y=10sin(pi/20,000(x-d))+70
How do I get d? Can someone clearly explain to me how to get the phase shift? thanks
make use of the fact that if
x = 2000 , y = 80
80 = 10sin(pi/20000(2000-d))+70
10 = 10sin(pi/20000(2000-d))
1 = sin(pi/20000(2000-d))
so (pi/20000(2000-d)) = pi/2
(2000-d)/10000 = 1
2000-d = 10000
d = -8000
so your equation is
y=10sin(pi/20000(x+8000))+70
(check: sub in x = 2000, it works
y = 80)
When I sub in x=2000, I get y=82.88 on my calculator?
How do I solve it to get 80?
NVM, I got it
Thank you for your help!
To determine the phase shift in a sinusoidal expression, you need to identify the starting point or reference point of the pattern. In this case, the year 2000 is the reference point because the average temperature is at a high point of 80.
To find the phase shift, you need to determine how many years it takes for the temperature to go from the highest point (80) to the reference point (2000). In this case, the time it takes for the temperature to go from the highest to the reference point is 20,000 years.
The general form of a sinusoidal function is y = A sin(B(x - C)) + D, where A represents the amplitude, B represents the period (or frequency), C represents the phase shift, and D represents the vertical shift.
Substituting the values into the equation, we have:
y = 10sin(pi/20,000(x - C)) + 70
Since the temperature at the reference point (2000) is at a high point, we want the phase shift (C) to make the expression evaluate to 0 at the reference point.
To find C, you can set up an equation and solve for it:
pi/20,000(2000 - C) = 0
By solving this equation, you can find the value of C, which represents the phase shift. Once you have C, you can substitute it back into the original equation to obtain the complete sinusoidal expression that models the phenomenon.
Note: Make sure to convert the units consistently (e.g., years for time) before performing any calculations, and consider rounding the result appropriately based on the desired precision.