Starting at c grams, the amount of radium-226 in a sample after years is given by:
 
 
 
 
 
**a)**      Rewrite the equation in the form, where r is the annual decay rate correct to four decimal places.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
**b)**     What is the annual decay rate as a percentage correct to 2 decimal places?
 
 
 
 

Given the original equation for radium-226 decay:

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

where:
- \( A(t) \) is the amount of radium-226 remaining after \( t \) years,
- \( c \) is the initial amount of radium-226,
- \( k \) is the decay constant, and
- \( t \) is the time in years.

We are asked to:
- **(a)** Rewrite the equation in the form \( A(t) = c \cdot (1 - r)^t \), where \( r \) is the annual decay rate.
- **(b)** Find the annual decay rate \( r \) as a percentage, correct to two decimal places.

### **Step (a): Rewriting in the form \( A(t) = c \cdot (1 - r)^t \)**

We need to convert the exponential form \( A(t) = c \cdot e^{-kt} \) into the form \( A(t) = c \cdot (1 - r)^t \).

To do this, we need to relate the two expressions:

1. \( e^{-k} \) represents the fraction of radium-226 remaining after 1 year.
2. Therefore, \( 1 - r = e^{-k} \), where \( r \) is the annual decay rate.
3. This leads to:
   \[
   r = 1 - e^{-k}
   \]

### **Step (b): Finding the annual decay rate as a percentage**

Once we have the value of \( r \), we can express it as a percentage by multiplying it by 100.

However, we need the value of \( k \), which depends on the half-life of radium-226. The half-life of radium-226 is approximately **1600 years**. Using this, we can compute \( k \) from the half-life formula:

\[
k = \frac{\ln(2)}{1600}
\]

Let me calculate this value and find \( r \).### (a) The decay constant \( k \) is approximately \( 0.0004332 \). Thus, the equation can be rewritten as:

\[
A(t) = c \cdot (1 - 0.0004331)^t
\]

where \( r = 0.0004331 \) is the annual decay rate.

### (b) The annual decay rate as a percentage is approximately **0.04%** (0.0433% when rounded to four decimal places).

Would you like further details or have any questions?

Here are five related questions for practice:

1. What would be the annual decay rate for a different element with a half-life of 2000 years?
2. How much radium-226 remains after 3200 years if you start with 100 grams?
3. If the initial amount of radium-226 is 50 grams, how much will remain after 800 years?
4. What is the decay constant for an element with a half-life of 500 years?
5. Can you derive the half-life formula from the exponential decay equation?

**Tip:** Remember that the decay rate \( r \) is always related to the half-life by the formula \( r = 1 - e^{-k} \), where \( k = \ln(2)/\text{half-life} \).

Starting at c grams, the amount of radium-226 in a sample after years is given by: a) Rewrite the equation in the form, where r is the annual decay rate correct to four decimal places. b) What is the annual decay rate as a percentage correct to 2 decimal places?

Explore how to rewrite the radium-226 decay equation to find the annual decay rate and its percentage. Learn about exponential decay and half-life formulas. This problem is ideal for high school students, involving concepts like logarithms and the exponential decay law.

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