AP Precalculus Practice Test: Unit 2 Question #13 TI-84+ to Use the Halflife Formula

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### AP Precalculus Practice Test: Unit 2, Question #13
**Topic:** Using the Half-Life Formula

This question involves applying the half-life formula to solve problems related to exponential decay. Half-life is a concept often used in chemistry, physics, and biology, where the quantity of a substance decreases by half over a specific period of time.

---

### **Half-Life Formula:**

The general formula used for half-life problems is:

\[
A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

Where:
- \( A(t) \) is the amount of substance remaining after time \( t \),
- \( A_0 \) is the initial amount of the substance,
- \( t \) is the time that has passed,
- \( T \) is the half-life of the substance (the time it takes for half of the substance to decay),
- \( \frac{1}{2} \) represents the halving of the substance in each period of the half-life.

---

### **Step-by-Step Explanation:**

1. **Identify Given Information:**
Typically, you will be given:
- The initial amount of the substance (\(A_0\)),
- The half-life (\(T\)),
- The amount remaining after a certain time (\(A(t)\)) or the time after which a specific amount remains.

2. **Substitute the Known Values into the Formula:**
Plug in the given values for \(A_0\), \(T\), and \(t\) into the half-life formula.

3. **Solve for the Unknown:**
Depending on what the problem asks, you may need to solve for \(A(t)\) (the remaining amount of the substance) or \(t\) (the time it takes for a given amount to decay).

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### **Example Problem:**

Suppose a substance has a half-life of 10 years. Initially, there are 100 grams of the substance. How much of the substance remains after 30 years?

### **Solution:**

1. **Identify the given values:**
- Initial amount, \( A_0 = 100 \) grams,
- Half-life, \( T = 10 \) years,
- Time elapsed, \( t = 30 \) years.

2. **Substitute into the half-life formula:**

\[
A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

\[
A(30) = 100 \left( \frac{1}{2} \right)^{\frac{30}{10}}
\]

Simplify the exponent:

\[
A(30) = 100 \left( \frac{1}{2} \right)^3 = 100 \cdot \frac{1}{8} = 12.5 \text{ grams}
\]

### **Final Answer:**

After 30 years, 12.5 grams of the substance remain.

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### **Practice Problem for Review:**

The half-life of a certain radioactive isotope is 5 years. If you start with 500 grams of the isotope, how much will remain after 20 years?

### **Solution for the Practice Problem:**

1. **Identify the given values:**
- Initial amount, \( A_0 = 500 \) grams,
- Half-life, \( T = 5 \) years,
- Time elapsed, \( t = 20 \) years.

2. **Substitute into the half-life formula:**

\[
A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

\[
A(20) = 500 \left( \frac{1}{2} \right)^{\frac{20}{5}} = 500 \left( \frac{1}{2} \right)^4 = 500 \cdot \frac{1}{16} = 31.25 \text{ grams}
\]

---

**Final Answer for Practice Problem:**
After 20 years, 31.25 grams of the isotope remain.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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