Nuclei of radioactive element `A` are produced at rate `t^(2)` (where `t` is time) at any time `t`. The element `A` has decay constant `lambda`. Let `N` be the number of nuclei of element `A` at any time `t`. At time `t=t_(0), dN//dt` is minimum. The number of nuclei of element `A` at time `t=t_(0)` is

Options

A. `(t_0-lambdat_0^2)/(lambda^2)`
B. `(2t_0-lambdat_0^2)/(lambda^2)`
C. `(t_0-lambdat_0^2)/(lambda)`
D. `(2t_0-lambdat_0^2)/(lambda)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Production and Decay of Nuclei
The number of nuclei \( N \) of the radioactive element \( A \) changes over time due to production and decay. The rate of production is given as \( t^2 \), and the decay is governed by the decay constant \( \lambda \). Therefore, the change in the number of nuclei can be expressed as:
\[ \frac{dN}{dt} = t^2 - \lambda N \]

### Step 2: Find the Condition for Minimum Rate of Change
We are given that at time \( t = t_0 \), the rate of change \( \frac{dN}{dt} \) is minimum. For this to happen, the second derivative of \( N \) with respect to time must be zero:
\[ \frac{d^2N}{dt^2} = 0 \]

### Step 3: Differentiate the Rate of Change Equation
Differentiating the equation \( \frac{dN}{dt} = t^2 - \lambda N \) with respect to time gives:
\[ \frac{d^2N}{dt^2} = 2t - \lambda \frac{dN}{dt} \]

### Step 4: Set the Second Derivative to Zero
Setting the second derivative to zero at \( t = t_0 \):
\[ 0 = 2t_0 - \lambda \frac{dN}{dt} \text{ at } t = t_0 \]
From the first equation, substituting \( t = t_0 \):
\[ \frac{dN}{dt} = t_0^2 - \lambda N \]

### Step 5: Substitute and Solve for \( N \)
Substituting \( \frac{dN}{dt} \) into the equation \( 0 = 2t_0 - \lambda (t_0^2 - \lambda N) \):
\[ 0 = 2t_0 - \lambda t_0^2 + \lambda^2 N \]
Rearranging gives:
\[ \lambda^2 N = 2t_0 - \lambda t_0^2 \]
Thus, we can express \( N \) as:
\[ N = \frac{2t_0 - \lambda t_0^2}{\lambda^2} \]

### Final Answer
The number of nuclei of element \( A \) at time \( t = t_0 \) is:
\[ N = \frac{2t_0 - \lambda t_0^2}{\lambda^2} \]