Example 10 from Precalculus, 7.6 Modeling with Trigonometric Functions

Finding a Cosine Function that Models Damped Harmonic Motion

Find and graph a function of the form $y = a e^{- c t} \cos (ω t)$ that models the information given.

ⓐ $a = 20, c = 0.05, p = 4$
ⓑ $a = 2, c = 1.5, f = 3$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
Substitute the given values into the model. Recall that period is $\frac{2 π}{ω}$ and frequency is $\frac{ω}{2 π} .$
1. ⓐ $y = 20 e^{- 0.05 t} \cos (\frac{π}{2} t) .$ See Figure 17.
  
  Figure 17
2. ⓑ $y = 2 e^{- 1.5 t} \cos (6 π t) .$ See Figure 18.
  
  Figure 18

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The following equation represents a damped harmonic motion model: $f (t) = 5 e^{- 6 t} \cos (4 t)$ Find the initial displacement, the damping constant, and the frequency.

How did it go?

Solution

Your turn

Solution

Your turn