Question
Determine whether the following represents continuous growth, continuous decay, or neither. \( y=3.5(e)^{-5 t} \)
Ask by Wagner Bowman. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The function \( y = 3.5e^{-5t} \) represents **continuous decay**.
Solution
The function given is:
\[ y = 3.5e^{-5t} \]
This is an exponential function of the form:
\[ y = Ce^{kt} \]
where:
- \( C = 3.5 \) is the initial value,
- \( k = -5 \) is the growth (if positive) or decay (if negative) rate.
### Analyzing the Exponent \( k \):
- **If \( k > 0 \)**: The function represents **continuous growth** because the exponent is positive, causing the function to increase exponentially as \( t \) increases.
- **If \( k < 0 \)**: The function represents **continuous decay** because the exponent is negative, causing the function to decrease exponentially as \( t \) increases.
- **If \( k = 0 \)**: The function is constant, neither growing nor decaying.
In the given function:
\[ k = -5 \]
Since \( k \) is negative, the function represents **continuous decay**.
### Conclusion:
The function \( y = 3.5e^{-5t} \) describes continuous decay.
**Answer:** It describes a situation of continuous exponential decay.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
The expression \( y=3.5(e)^{-5 t} \) represents continuous decay. This is because the function involves an exponential term with a negative exponent, indicating that as time \( t \) increases, the value of \( y \) will decrease exponentially. In continuous growth, the exponential function would have a positive exponent, leading to an increase in \( y \) as \( t \) increases. So, in this case, the negative exponent reflects the decay of the quantity over time.
