7.6 Modeling with trigonometric equations  (Page 5/14)

Substitute the given values into the model. Recall that period is $\frac{2\pi }{\omega }$ and frequency is $\frac{\omega }{2\pi }.$

1. $y=20e^{-0.05t}\cos \left(\frac{\pi }{2}t\right).$ See [link] .

   

2. $y=2e^{-1.5t}\cos \left(6\pi t\right).$ See [link] .

   

Calculate the value of $\omega$ and substitute the known values into the model.

1. As period is $\frac{2\pi }{\omega },$ we have
   $$\begin{array}{l}\frac{\pi }{6}=\frac{2\pi }{\omega }\hfill \\ \omega \pi =6(2\pi )\hfill \\ \omega =12\hfill \end{array}$$

   The damping factor is given as 10 and the amplitude is 7. Thus, the model is $y=7e^{-10t}\sin \left(12t\right).$ See [link] .

   

2. As frequency is $\frac{\omega }{2\pi },$ we have
   $$\begin{array}{l}20=\frac{\omega }{2\pi }\hfill \\ 40\pi =\omega \hfill \end{array}$$

   The damping factor is given as $0.2$ and the amplitude is $\mathrm{0.3.}$ The model is $y=0.3e^{-0.2t}\sin \left(40\pi t\right).$ See [link] .

   

The amplitude begins at 5 in. and deceases 30% each second. Because the spring is initially compressed, we will write A as a negative value. We can write the amplitude portion of the function as

We put ${\left(1-0.30\right)}^t$ in the form $e^{ct}$ as follows:

Now let’s address the period. The spring cycles through its positions every 3 seconds, this is the period, and we can use the formula to find omega.

The natural length of 10 inches is the midline. We will use the cosine function, since the spring starts out at its maximum displacement. This portion of the equation is represented as

Finally, we put both functions together. Our the model for the position of the spring at $t$ seconds is given as

See the graph in [link] .

The displacement factor represents the amplitude and is determined by the coefficient $a{e}^{-ct}$ in the model for damped harmonic motion. The damping constant is included in the term $e^{-ct}.$ It is known that after 3 seconds, the local maximum measures one-half of its original value. Therefore, we have the equation

Use algebra and the laws of exponents to solve for $c.$

Then use the laws of logarithms.

The damping constant is $\frac{\ln 2}{3}.$