A radioactive substance decays exponentially. A scientist begins with 180 milligrams of a radioactive substance. After 29 hours, 90 mg of the substance remains.

How many milligrams will remain after 51 hours

Give your answer accurate to at least one decimal place.

The decay of a radioactive substance is modeled by the exponential decay formula:

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

Where:
- \( A(t) \) is the amount of the substance at time \( t \),
- \( A_0 \) is the initial amount of the substance,
- \( k \) is the decay constant,
- \( t \) is the time that has passed.

### Step 1: Determine the decay constant \( k \)
We know that:
- \( A_0 = 180 \) mg,
- After 29 hours, \( A(29) = 90 \) mg.

Using the decay formula:
\[
90 = 180 \cdot e^{-29k}
\]

Dividing both sides by 180:
\[
\frac{90}{180} = e^{-29k}
\]
\[
0.5 = e^{-29k}
\]

Taking the natural logarithm (ln) of both sides:
\[
\ln(0.5) = -29k
\]
\[
k = \frac{-\ln(0.5)}{29}
\]

Now, calculate \( k \).

\[
k = \frac{-\ln(0.5)}{29} \approx \frac{0.6931}{29} \approx 0.0239
\]

### Step 2: Calculate the remaining amount after 51 hours
Now that we know \( k \approx 0.0239 \), we can find the remaining amount after 51 hours.

Using the decay formula again:
\[
A(51) = 180 \cdot e^{-0.0239 \cdot 51}
\]

Let's calculate this value.

\[
A(51) \approx 180 \cdot e^{-1.219}
\]
\[
A(51) \approx 180 \cdot 0.295
\]
\[
A(51) \approx 53.1 \, \text{mg}
\]

### Final Answer:
After 51 hours, approximately **53.1 mg** of the substance will remain.

Would you like any further explanation on the steps?

Here are 5 related questions:
1. How is the exponential decay formula derived?
2. What is the half-life of this radioactive substance?
3. How can exponential decay apply to other fields besides radioactivity?
4. How does the value of \( k \) affect the rate of decay?
5. Can the exponential decay formula be solved graphically?

**Tip:** In exponential decay, the rate of decay is proportional to the current amount of the substance.

A radioactive substance decays exponentially. A scientist begins with 180 milligrams of a radioactive substance. After 29 hours, 90 mg of the substance remains. How many milligrams will remain after 51 hours? Give your answer accurate to at least one decimal place.

Learn how to calculate the remaining amount of a radioactive substance using exponential decay. Starting with 180 mg, after 29 hours only 90 mg remains. Find out how many milligrams will remain after 51 hours with a detailed step-by-step solution. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation