When a car hits a certain bump on the road, a shock absorber on the car is compressed a distance of 8 in., then released. The shock absorber vibrates in damped harmonic motion with a frequency of 6 cycles per second. The damping constant for this particular shock absorber is 3.4.
I found the equation of f(t)=8e^-3.4tcos(12πt).
How long does it take for the amplitude of the vibration to decrease to 0.8 in.? (Round your answer to two decimal places.)

Question

When a car hits a certain bump on the road, a shock absorber on the car is compressed a distance of 8 in., then released. The shock absorber vibrates in damped harmonic motion with a frequency of 6 cycles per second. The damping constant for this particular shock absorber is 3.4.
I found the equation of f(t)=8e^-3.4tcos(12πt).
How long does it take for the amplitude of the vibration to decrease to 0.8 in.? (Round your answer to two decimal places.)

aum · Answer

Assuming the equation for f(t) is correct, the amplitude is $\large 8e^{-3.4t}$.
Equate the amplitude to 0.8 and solve for t.

anonymous · Answer

The equation for f(t) is correct but I'm stuck on trying to solve for t. My professor never went over the chapter that this is in.

aum · Answer

$$
8e^{-3.4t} = 0.8 \
\text{Divide by 8:} \
e^{-3.4t} = 0.1 \
\text{Take ln on both sides:} \
-3.4t = \ln(0.1) \
t = \frac{\ln(0.1)}{-3.4}  = ?
$$

anonymous · Answer

Ah, thank you!