If 

1616 

g of a radioactive substance are present initially and 

88 

yr later only 

8.08.0 

g​ remain, how much of the​ substance, to the nearest tenth of a​ gram, will be present after 

1111 

​yr?

Question content area bottom

Part 1

After 

1111 

​yr, there will be 

enter your response here 

g of the radioactive substance.

To solve this problem, we need to use the formula for exponential decay of a radioactive substance:

\[ N(t) = N_0 e^{-kt} \]

where:
- \( N(t) \) is the remaining amount of substance at time \( t \)
- \( N_0 \) is the initial amount of the substance
- \( k \) is the decay constant
- \( t \) is the time in years

First, we need to find the decay constant \( k \). We know that initially \( N_0 = 1616 \) grams, and after 88 years \( N(88) = 8.0 \) grams. Substituting these values into the formula, we get:

\[ 8.0 = 1616 e^{-88k} \]

We can solve for \( k \) by dividing both sides by 1616:

\[ \frac{8.0}{1616} = e^{-88k} \]

\[ \frac{1}{202} = e^{-88k} \]

Taking the natural logarithm of both sides:

\[ \ln\left(\frac{1}{202}\right) = -88k \]

\[ -\ln(202) = -88k \]

\[ k = \frac{\ln(202)}{88} \]

Using a calculator to find the natural logarithm of 202:

\[ \ln(202) \approx 5.3083 \]

So,

\[ k = \frac{5.3083}{88} \approx 0.0603 \]

Now that we have the decay constant \( k \), we can use it to find the amount of substance remaining after 111 years. We use the formula again with \( t = 111 \):

\[ N(111) = 1616 e^{-0.0603 \times 111} \]

Calculating the exponent:

\[ -0.0603 \times 111 \approx -6.6993 \]

So,

\[ N(111) = 1616 e^{-6.6993} \]

Using a calculator to find \( e^{-6.6993} \):

\[ e^{-6.6993} \approx 0.0012 \]

Thus,

\[ N(111) = 1616 \times 0.0012 \approx 1.9 \]

To the nearest tenth of a gram, there will be approximately 1.9 grams of the radioactive substance remaining after 111 years.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. What is the half-life of the radioactive substance based on the given decay constant?
2. How would you find the amount of substance remaining after a different number of years?
3. What is the significance of the decay constant in the context of radioactive decay?
4. How does changing the initial amount of substance affect the amount remaining after a given time?
5. Can you derive the exponential decay formula from basic principles?
6. What are some real-world applications of exponential decay in radioactive substances?
7. How does the decay constant relate to the stability of a radioactive substance?
8. What is the mathematical relationship between the decay constant and the half-life of a substance?

**Tip:** Always double-check your calculations when working with exponential functions and logarithms to ensure accuracy.

Learn how to calculate the amount of a radioactive substance remaining after 111 years using the exponential decay formula. This problem involves initial substance amount, decay constant, and time calculation. Suitable for advanced high school students.

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