Set Up an Equation That Models Harmonic Motion - Pre-Calculus

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Question

Create an equation modelling temperature , with highest temperature at , which is degrees and lowest temperature of degrees which occurs at . Assume that this model is sinusoidal and use a cosine model.

Answer

This can be written in the general form of:

.

Since the maximum occurs at , we can arbitrarily choose since cosine would be maximum when the inner term is equal to .

To determine , let's determine the period first.

The period is equal to twice the length between adjacent crest and trough.

For us that is:

To determine , we do

$b=\frac{2\pi}{T}=\frac{2\pi}{24}=\frac{\pi}{12}$

To determine ,

$A=\frac{(max-min)}{2}=\frac{85-65}{2}=\frac{20}{2}=10$

To determine ,

$D=\frac{(max+min)}{2}=\frac{65+85}{2}=\frac{150}{2}=75$

The entire regression can therefore be written as:

$y(t)=10cos(\frac{\pi}{12}(t-12))+75$

The only thing that can be changed to keep the regression the same is the phase shift , and sign of the amplitude . The other two terms must be kept as they are.

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Question

Create an equation modelling temperature , with highest temperature at , which is degrees and lowest temperature of degrees which occurs at . Assume that this model is sinusoidal and use a cosine model.

Answer

This can be written in the general form of:

.

Since the maximum occurs at , we can arbitrarily choose since cosine would be maximum when the inner term is equal to .

To determine , let's determine the period first.

The period is equal to twice the length between adjacent crest and trough.

For us that is:

To determine , we do

$b=\frac{2\pi}{T}=\frac{2\pi}{24}=\frac{\pi}{12}$

To determine ,

$A=\frac{(max-min)}{2}=\frac{85-65}{2}=\frac{20}{2}=10$

To determine ,

$D=\frac{(max+min)}{2}=\frac{65+85}{2}=\frac{150}{2}=75$

The entire regression can therefore be written as:

$y(t)=10cos(\frac{\pi}{12}(t-12))+75$

The only thing that can be changed to keep the regression the same is the phase shift , and sign of the amplitude . The other two terms must be kept as they are.

Compare your answer with the correct one above

Question

Create an equation modelling temperature , with highest temperature at , which is degrees and lowest temperature of degrees which occurs at . Assume that this model is sinusoidal and use a cosine model.

Answer

This can be written in the general form of:

.

Since the maximum occurs at , we can arbitrarily choose since cosine would be maximum when the inner term is equal to .

To determine , let's determine the period first.

The period is equal to twice the length between adjacent crest and trough.

For us that is:

To determine , we do

$b=\frac{2\pi}{T}=\frac{2\pi}{24}=\frac{\pi}{12}$

To determine ,

$A=\frac{(max-min)}{2}=\frac{85-65}{2}=\frac{20}{2}=10$

To determine ,

$D=\frac{(max+min)}{2}=\frac{65+85}{2}=\frac{150}{2}=75$

The entire regression can therefore be written as:

$y(t)=10cos(\frac{\pi}{12}(t-12))+75$

The only thing that can be changed to keep the regression the same is the phase shift , and sign of the amplitude . The other two terms must be kept as they are.

Compare your answer with the correct one above

Question

Create an equation modelling temperature , with highest temperature at , which is degrees and lowest temperature of degrees which occurs at . Assume that this model is sinusoidal and use a cosine model.

Answer

This can be written in the general form of:

.

Since the maximum occurs at , we can arbitrarily choose since cosine would be maximum when the inner term is equal to .

To determine , let's determine the period first.

The period is equal to twice the length between adjacent crest and trough.

For us that is:

To determine , we do

$b=\frac{2\pi}{T}=\frac{2\pi}{24}=\frac{\pi}{12}$

To determine ,

$A=\frac{(max-min)}{2}=\frac{85-65}{2}=\frac{20}{2}=10$

To determine ,

$D=\frac{(max+min)}{2}=\frac{65+85}{2}=\frac{150}{2}=75$

The entire regression can therefore be written as:

$y(t)=10cos(\frac{\pi}{12}(t-12))+75$

The only thing that can be changed to keep the regression the same is the phase shift , and sign of the amplitude . The other two terms must be kept as they are.

Compare your answer with the correct one above

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