Initial activity of the radioactive substance = ?

Given data:
- The substance is 300 days old.
- Its current activity is 5000 dps.
- 150 days later, its activity becomes 2500 dps.

To find the initial activity of the substance, we need to consider its radioactive decay over time.

1. Understanding Radioactive Decay:
Radioactive decay is a process in which the number of radioactive atoms decreases over time, leading to a reduction in the activity of the substance. The rate of decay is given by the decay constant (λ), which is specific to each radioactive material.

2. Using the Decay Law:
The decay law states that the rate of decay of a radioactive substance is proportional to its current activity. Mathematically, it can be expressed as:

A(t) = A₀ * e^(-λt)

Where:
- A(t) is the current activity at time t
- A₀ is the initial activity (unknown)
- λ is the decay constant
- t is the time passed

3. Applying the Given Data:
We are given two pieces of information:
- The substance is 300 days old, and its activity is 5000 dps.
- 150 days later, its activity becomes 2500 dps.

Using the decay law, we can set up two equations:

5000 = A₀ * e^(-λ * 300) [Equation 1]
2500 = A₀ * e^(-λ * 450) [Equation 2]

4. Solving the Equations:
To solve these equations, we can divide Equation 2 by Equation 1:

2500/5000 = e^(-λ * 450) / e^(-λ * 300)

0.5 = e^(-λ * 150)

Taking the natural logarithm of both sides:

ln(0.5) = -λ * 150

Solving for λ:

λ = -ln(0.5) / 150

Substituting the value of λ back into Equation 1:

5000 = A₀ * e^(ln(0.5) / 150 * 300)

Simplifying:

5000 = A₀ * 0.5^(2)

5000 = A₀ * 0.25

Dividing both sides by 0.25:

A₀ = 5000 / 0.25

A₀ = 20000 dps

Therefore, the initial activity of the radioactive substance was 20000 dps (option B).