To analyze the function \( y = 80(0.69)^{5t} \), we need to identify whether it represents growth or decay.
The base of the exponential expression is \(0.69\), which is less than 1. This indicates that the function represents a decay process.
To find the constant percent rate of decay, we can apply the formula for calculating the decay rate:
\[ \text{Decay rate} = 1 - \text{base} = 1 - 0.69 = 0.31 \]
This means the function is decreasing by 31% for each unit increase in \( t \). To express this as a percent rate of decay:
\[ \text{Rate of decay} = 31% \text{ per } 5t \]
However, to find the equivalent decay rate for \( t \):
The decay rate for \( t \) is calculated as:
\[ \text{Percent Rate} = 31% \div 5 \approx 6.2% \text{ per unit of } t \]
Since you are looking for options given in decay rates, there seems to be a misunderstanding in the available choices since the calculated rate does not directly correspond with the listed options.
Given your options:
- The function has a constant percent rate of decay of about 16%.
- The function has a constant percent rate of decay of about 84%.
- Etc.
None of the above options correlate correctly with our results. The computed decay percentage from \(0.69\) is around 31% and not listed as such. The statement that best matches based on the given options would primarily revolve around an identified decay that is characteristic of expiring at a sub 1% ratio.
The closest choice amongst those previously stated regarding a slowing rate would be The function has a constant percent rate of decay of about 16%., if we consider possible logistical adjustments but the accurate decay rate is more extensively breakdown at about 31%.
Summarizing, the closest answer based on those listings is towards the notion of decay, despite an actual numerical mismatch. Would you like to verify this in a specific manner?
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