A specific radioactive substance follows a continuous exponential decay model. It has a half-life of  4 hours. At the start of the experiment, 89.8g is present. How much will be present in 21 hours?
Do not round any intermediate computations, and round your answer to the nearest tenth.

To solve this problem, we need to use the exponential decay formula, which is given by:

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

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

### Step 1: Determine the decay constant \( k \)
The decay constant \( k \) is related to the half-life (\( t_{1/2} \)) by the formula:

\[
k = \frac{\ln(2)}{t_{1/2}}
\]

Given the half-life \( t_{1/2} = 4 \) hours, we can calculate \( k \):

\[
k = \frac{\ln(2)}{4} = \frac{0.693147}{4} \approx 0.1732867 \, \text{hours}^{-1}
\]

### Step 2: Apply the exponential decay formula
Now that we have \( k \), we can substitute the values into the exponential decay formula to find the amount of substance remaining after 21 hours.

\[
A(21) = 89.8 \cdot e^{(0.1732867 \times -21)}
\]

### Step 3: Compute the exponent and the remaining amount
First, compute the exponent:

\[
0.1732867 \times -21 = -3.6380207
\]

Then, calculate \( e^{-3.6380207} \):

\[
e^{-3.6380207} \approx 0.026211
\]

Finally, multiply by the initial amount \( A_0 = 89.8 \):

\[
A(21) \approx 89.8 \times 0.026211 \approx 2.353208
\]

### Step 4: Round the result
Rounding to the nearest tenth:

\[
A(21) \approx 2.4 \, \text{g}
\]

Thus, after 21 hours, approximately **2.4 grams** of the substance will be present.

Would you like any further details or have any questions? Here are some related questions you might consider:

1. How would the remaining amount change if the half-life was 2 hours instead of 4?
2. What would happen if the experiment lasted for 30 hours instead of 21?
3. How can we find the time required for the substance to decay to a specific amount?
4. What does the decay constant \( k \) represent in practical terms?
5. How would the equation change if the substance followed a different decay model?
6. What is the importance of not rounding intermediate steps in this kind of calculation?
7. How would the calculations differ if the initial amount were different, say 100g?
8. Can you derive the formula for the half-life from the exponential decay equation?

**Tip:** When dealing with exponential decay, always ensure to handle the decay constant carefully, as small errors in \( k \) can lead to significant differences in the final result.

Learn how to calculate the remaining amount of a radioactive substance using an exponential decay model with a half-life of 4 hours. Start with 89.8 grams and find out how much will be present after 21 hours. Detailed step-by-step solution provided.

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