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?

Question

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

Reiny · Answer

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)

DK · Answer

When I sub in x=2000, I get y=82.88 on my calculator? How do I solve it to get 80?

DK · Answer

NVM, I got it Thank you for your help!