The the product of the initial amplitude and the exponential decay rate
of raised to the power of the time taken give the amplitude of 0.1 mm.
Correct option:
- The amplitude will reach 0.1 mm in about 7 seconds
What is the exponential decay rate and how is it used?
The given parameters are;
The initial amplitude = 0.75 mm
The decay rate, i = 25%
Required:
The time it takes the guitar to decay to 0.1 mm.
Solution:
The series of the sizes of the amplitude that has constant decay uses an
exponential decay formula.
When the amplitude is 0.1 mm, we have;
y = a·rⁿ
Where;
r = 1 - i
a = 0.75
n = The time taken
Which gives;
r = 1 - 0.25 = 0.75
y = 0.1 = 0.75 × 0.75ⁿ
[tex]\dfrac{0.1}{0.75} = \mathbf{0.75^{n}}[/tex]
- [tex]n = \dfrac{ln\left( \dfrac{0.1}{0.75} \right)}{ln(0.75)} \approx \mathbf{7.0}[/tex]
n = 7.0
- The time it takes the amplitude to reach 0.1 mm is 7 seconds.
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