Question
Determine whether the following represents continuous growth, continuous decay, or neither. \( y=3.5(e)^{-5 t} \)
Ask by Stuart Potter. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The function \( y = 3.5(e)^{-5t} \) represents **continuous decay**.
Solution
To determine whether the function \( y = 3.5(e)^{-5t} \) represents continuous growth, continuous decay, or neither, we can analyze the exponent of the exponential function.
1. **Identify the form of the function**: The function is in the form \( y = a e^{kt} \), where:
- \( a = 3.5 \) (the initial value),
- \( k = -5 \) (the growth/decay rate),
- \( t \) is the time variable.
2. **Analyze the value of \( k \)**:
- If \( k > 0 \), the function represents continuous growth.
- If \( k < 0 \), the function represents continuous decay.
- If \( k = 0 \), the function is constant.
In this case, since \( k = -5 \) (which is less than 0), the function represents continuous decay.
Thus, the function \( y = 3.5(e)^{-5t} \) represents **continuous decay**.
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Simplify this solution Beyond the Answer
The equation \( y=3.5(e)^{-5 t} \) represents continuous decay. This is evident because the exponent of \( e \) is negative (-5t), which indicates that the quantity decreases as time \( t \) increases. In continuous decay models, we typically see a constant reduction over time rather than growth. You can find applications of continuous decay in various fields, such as biology, where it can describe population decline, and in physics, where it can explain radioactive decay. For example, if a certain substance has a half-life, knowing its continuous decay formula can help predict how much will remain after specific intervals.
