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When discussing radioactive decay, one frequently encounters the concept of "half-life." Half-life is the time it takes for a substance to decrease to half its initial amount. Consider it as a clock that counts down to half the original sample size. To calculate the half-life of a substance, you first need the decay constant, a value integral to determining how quickly the substance decays.
In the given problem, we determined the decay constant, and with this, the half-life was calculated. You can use the formula:

• Half-life formula: \( t_{1/2} = \frac{\ln 2}{k} \)

The natural logarithm of 2, \( \ln 2 \), approximately equals 0.693. Plug this into the formula, and you can find the time it will take for the substance to decay to half its original amount. It’s like understanding how long food stays fresh but in terms of radioactive matter.

The decay constant, \( k \), is a critical value in understanding radioactive decay. This constant describes the rate at which a radioactive substance decays. In simpler terms, it tells you how quickly you can expect a substance to decrease over time.
To find \( k \), you often need to rearrange the exponential decay formula:

• Start with the formula: \( A(t) = A_0 e^{-kt} \)
• Plug in the known values for \( A(t) \), \( A_0 \), and \( t \).
• Example: \( 3.5 = 10 e^{-5k} \)
• Divide to simplify, then take the natural logarithm: \( \ln(0.35) = -5k \)
• Solve for \( k \): \( k = -\frac{\ln(0.35)}{5} \)

Here, \( k \approx 0.20996 \). This value ultimately helps in determining how fast the material is decaying, and is a stepping stone to finding the half-life.

At the heart of this problem is the radioactive decay formula, which models how a substance decreases over time. The standard formula in use is the exponential decay formula:

• \( A(t) = A_0 e^{-kt} \)

This formula consists of several parts:

• \( A(t) \): the amount remaining after time \( t \)
• \( A_0 \): the initial quantity of the substance
• \( e \): the base of the natural logarithm, approximately 2.71828
• \( k \): the decay constant

In practical terms, by substituting the specific values into this equation, a scientist can predict how much of a substance will remain after a certain time period. In our example, after 5 days, the substance reduced from 10.0 mg to 3.5 mg. Using this model, you can pinpoint how much will be left at future times, or work backwards to find out past quantities.