Question

Outside temperature over a day can be modelled using a sine or cosine function. Suppose you know the high temperature for the day is 72 degrees and the low temperature of 62 degrees occurs at 3 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer 1

The equation which represents the equation for the temperature, D , in terms of t is D(t)=5°cos{(π/3)t}+67°.

Given that the high temperature is 72 degrees and low temperature is 62 degrees at 3 A.M.

We know that temperature is the intensity of the heat present around us.

We know that,

Maximum temperature=72 degrees,

Minimum temperature=62 degrees, which occurs at t=3 hours

Now we can write the equation as:

D(t)=A cos(ct)+B

Where A, c, B are constants.

We have a minimum at t=3 a minimum means cos(ct)=-1

then we have that D(3)=A cos(c*3)+B

=A*(-1)+b

=35°

Here we solve that ,

Cos(c*3)=-1

this means that

c*3=-1

c*3=π

c=π/3

We also know that the maximum temperature is 72°, the maximum temperature is when cos(c*t)=1

D(t)=0=A(t)+B=72

With this we can find that values of A and b

-A+B=62

A+B=72

B=67

A=5

Equation will be D(t)=5 cos{(π/3)t}+67°.

Hence the equation for the temperature is D(t)=5 cos{(π/3)t}+67°..

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