The mass of a radioactive substance follows a *continuous exponential decay model*, with a decay rate parameter of 3.8% per day. Find the half-life of this substance (that is, the time it takes for one-half the original amount in a given sample of this substance to decay). **Note:** This is a *continuous* exponential decay model.

Do not round any intermediate computations, and round your answer to the nearest hundredth.


To solve for the half-life of a radioactive substance following a continuous exponential decay model, we'll use the formula for continuous decay:

\[
A(t) = A_0 \times e^{kt}
\]

where:
- \( A(t) \) is the amount of substance remaining at time \( t \),
- \( A_0 \) is the initial amount of substance,
- \( k \) is the decay constant, and
- \( t \) is the time.

### Step 1: Understand the problem
Given:
- Decay rate parameter \( r = 3.8\% \) per day, which we convert to a decimal: \( r = 0.038 \).

We need to find the half-life \( t_{1/2} \), which is the time when \( A(t) = \frac{1}{2} A_0 \).

### Step 2: Set up the equation for half-life
Using the decay formula, when \( A(t) = \frac{1}{2} A_0 \), we get:

\[
\frac{1}{2} A_0 = A_0 \times e^{kt_{1/2}}
\]

### Step 3: Simplify the equation
Dividing both sides by \( A_0 \):

\[
\frac{1}{2} = e^{kt_{1/2}}
\]

Taking the natural logarithm (ln) of both sides:

\[
\ln\left(\frac{1}{2}\right) = kt_{1/2}
\]

### Step 4: Substitute the decay constant
The decay constant \( k \) is related to the rate by \( k = -r \), so:

\[
\ln\left(\frac{1}{2}\right) = -0.038 \times t_{1/2}
\]

### Step 5: Solve for \( t_{1/2} \)
We will solve for \( t_{1/2} \) now.The half-life of the substance is approximately **18.24 days**.

Would you like more details, or do you have any questions?

Here are 8 related questions:
1. How would the half-life change if the decay rate was increased to 5% per day?
2. What is the formula for the decay constant \( k \) in exponential decay?
3. How do you calculate the remaining mass of a substance after a certain number of half-lives?
4. What is the difference between continuous and discrete exponential decay models?
5. How do you derive the half-life formula from the exponential decay model?
6. What is the half-life of a substance with a decay rate of 1% per hour?
7. How can you determine the decay rate if the half-life is known?
8. How does the concept of half-life apply to other fields such as finance or medicine?

**Tip:** When dealing with logarithmic equations, it is helpful to remember the properties of logarithms to simplify and solve the equation efficiently.

Learn how to calculate the half-life of a radioactive substance using a continuous exponential decay model with a decay rate of 3.8% per day. Understand the decay formula and solve for the half-life step-by-step. Suitable for advanced high school students and learners interested in physics and chemistry.

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