If `-((dN)/(dt))_(o)` is the initial activity and `-((dN)/(dt))` is the activity at time t in a radioactive disintegration then :

Options

A. `-((dN)/(dt))_(o)=-((dN)/(dt))e^(-kt)`
B. `-((dN)/(dt))_(o)=((dN)/(dt))_(o)e^(-kt)`
C. `-((dN)/(dt))_(o)=-((dN)/(dt))_(o)e^(-kt)`
D. `-((dN)/(dt))_(o)=((dN)/(dt))_(o)e^(-kt)`

Explanation

To solve the problem, we need to relate the initial activity of a radioactive substance to its activity at a later time using the principles of radioactive decay. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding Activity**:
The activity of a radioactive substance is defined as the rate of decay of the substance. Mathematically, it can be expressed as:
\[ -\frac{dN}{dt} = \lambda N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive nuclei at time \( t \).

2. **Initial Activity**:
At time \( t = 0 \), the initial number of nuclei is \( N_0 \). Therefore, the initial activity can be written as:
\[ -\frac{dN}{dt}\bigg|_{t=0} = \lambda N_0 \]

3. **Number of Nuclei at Time \( t \)**:
The number of radioactive nuclei at any time \( t \) can be expressed using the exponential decay formula:
\[ N = N_0 e^{-\lambda t} \]

4. **Activity at Time \( t \)**:
Substituting \( N \) into the activity equation gives us the activity at time \( t \):
\[ -\frac{dN}{dt} = \lambda N = \lambda (N_0 e^{-\lambda t}) = \lambda N_0 e^{-\lambda t} \]

5. **Relating Initial Activity to Activity at Time \( t \)**:
We can express the activity at time \( t \) in terms of the initial activity:
\[ -\frac{dN}{dt} = -\frac{dN}{dt}\bigg|_{t=0} e^{-\lambda t} \]
This shows that the activity at time \( t \) is equal to the initial activity multiplied by the exponential decay factor.

6. **Final Relationship**:
Therefore, we can conclude that:
\[ -\frac{dN}{dt} = -\frac{dN}{dt}\bigg|_{t=0} e^{-\lambda t} \]

### Conclusion:
The activity at time \( t \) is related to the initial activity by an exponential decay factor.