Problem 47 - Elementary and Intermediate Algebra

Question

Radioactive Decay. Five hundred grams of a radioactive material decays according to the formula \(A=500\left(\frac{2}{3}\right)^{t},\) where \(t\) is measured in years. Find the amount present in 10 years. Round to the nearest one-tenth of a gram.

Step-by-Step Solution

Answer

The amount present in 10 years is approximately 8.7 grams.

Step 1: Understand the given formula

The formula for radioactive decay is given by \( A = 500 \left( \frac{2}{3} \right)^{t} \). Here, \( A \) is the amount of substance present at time \( t \), \( 500 \) is the initial amount of substance, and \( \left( \frac{2}{3} \right)^{t} \) represents the decay over time. The value \( t \) is the time in years.

Step 2: Substitute the time value

We need to find the amount present after 10 years, which means we will substitute \( t = 10 \) into the formula. This gives:\[ A = 500 \left( \frac{2}{3} \right)^{10} \]

Step 3: Calculate the decay factor raised to the power of time

Calculate \( \left( \frac{2}{3} \right)^{10} \). This is a repeated multiplication:\[ \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) \times \cdots \text{(a total of 10 times)} \approx 0.01734 \]

Step 4: Multiply to find the remaining amount

Multiply the initial amount by the decay factor:\[ A = 500 \times 0.01734 \approx 8.67 \]

Step 5: Round the result

Round 8.67 to the nearest one-tenth of a gram, which gives us 8.7 grams.

Key Concepts

Exponential Decay Formula

The exponential decay formula is essential in understanding how quantities reduce over time. Specifically, for radioactive decay, the formula is expressed as \( A = A_0 imes \left( \frac{x}{y} \right)^t \).

• \( A \) represents the remaining amount of the radioactive material at time \( t \).
• \( A_0 \) is the initial quantity, in our case, 500 grams.
• The fraction \( \left( \frac{x}{y} \right) \) is the decay factor, which determines how quickly the material decays.
• \( t \) is the time period over which the decay occurs.

Understanding this formula is like having a tool that helps predict how much of a radioactive substance remains after some time. For any exponential decay situation, whether it's radioactive decay or another process like depreciation in economics, the principles remain the same.

Radioactive Material

Radioactive materials are substances that emit radiation as they break down or decay. This natural process involves the transformation of an unstable atomic nucleus.

• They can release different types of radiation, such as alpha, beta, or gamma rays.
• Each type of radioactive material decays at a specific rate, expressed mathematically by its half-life.
• The half-life is the time it takes for half of the material to decay, which is central to understanding how the amount decreases over time.

In our exercise, the material began at 500 grams, and through decay, the amount reduces in accordance with the decay formula. Knowing how radioactive materials behave helps in fields ranging from medicine to archaeology, where radioactive dating provides insights into the age of objects.

Decay Calculation

Calculating the decay involves determining how much of a radioactive material remains after a certain period. In the given problem, we used the formula \( A = 500 \left( \frac{2}{3} \right)^{10} \) to find this amount after 10 years.

• The decay factor \( \left( \frac{2}{3} \right) \) is raised to the power of \( t \), which in this case is 10.
• This computes the repeated multiplication, simulating the decay over the years. The result was approximately \( 0.01734 \).
• Finally, multiplying the original quantity (500 grams) by this value gave us \( A \approx 8.67 \).
• Rounding \( 8.67 \) to the nearest one-tenth of a gram resulted in \( 8.7 \) grams remaining.

Through this calculation, we could accurately assess the remaining radioactive material, highlighting the importance of precise arithmetic and understanding of exponential functions in decay processes.