Question

Scientists believe that the average temperatures at various places on Earth vary from cooler to warmer over thousands of years. At one place on Earth, the highest average temperature is 800 and the lowest is 600 . Suppose it takes 20,000 years to go from the high to low average and the average temperature was at a high point of 800 in year 2000. Set up a sinusoidal function T t( ) (where t is time in years) to model this phenomenon.

Answer 1

Answer:

$T(t)=100*sin(\frac{2\pi}{40000}*(t-2000)+\frac{\pi}{2})$

Explanation:

The cosinusoidal equation has a the following structure:

$T(t)=A*cos(\frac{2\pi}{P} (t-t_o))$

Where:

A: amplitude from the wave. If max =800 and min=600. Then 2*A=(max-min)=200. A=100

P: Period of the wave. If it takes 20000 years go from the high to the low. Then P/2=20000, P=40000

to: is the time at the time of a maximum. to=2000

Relation between sin and cos:

$cos(\alpha)=sin(\alpha+\pi/2)$

We replace all these values in the cosinus equation:

$T(t)=100*sin(\frac{2\pi}{40000}*(t-2000)+\frac{\pi}{2})$

Answer 2

Answer:

T (t) = A x cos (2π/40000 x (t - 2000) +π/2)

Explanation:

From the question given, we recall the following,

The co-sinusoidal equation formation is denoted  as:

T (t) = A x cos (2π/p (t - t₀))

At one place on earth the highest temperature = 800

At one place on earth  The lowest temperature = 600

A: represents the wave amplitude, if the max and min are both = 800 and 600  respectively.

Thus,

2 x A = maximum -minimum = 200,

A = 100

P: Represents the wave period

Now suppose it takes 20,000 years to go from the high to the low average

Then,

P/2=20000, P=40000

t₀: Represents  the time at the time of a maximum. t₀=2000

The Relationship  between cos and sin is:

cos (α)= sin ( α + π/2)

In the cosinus equation, all the values are replaced.

Therefore,

We have, T (t) = A x cos (2π/40000 x (t - 2000) +π/2)