Problem 85 - Beginning and Intermediate Algebra

Question

One gram of radium produces 37 billion disintegrations per second. How many disintegrations are produced in 1 hour?

Step-by-Step Solution

Answer

One gram of radium produces 37 billion disintegrations per second. To find the total disintegrations in 1 hour, we convert 1 hour to seconds (1 hour = 60 × 60 seconds), and then multiply the given rate by the total seconds in an hour: \(37,000,000,000 \times 3,600 = 133,200,000,000,000\) disintegrations.

Step 1: Convert hours to seconds

First, we need to convert 1 hour into seconds. We know that 1 hour = 60 minutes and 1 minute = 60 seconds. So, 1 hour = 60 × 60 seconds.

Step 2: Calculate the total number of disintegrations in 1 hour

We are given that there are 37 billion disintegrations per second. Let's multiply this by the total seconds found in Step 1 to get the number of disintegrations in one hour. Total disintegrations = disintegrations per second × total seconds in an hour

Step 3: Perform the calculation

Now, plug in the values and perform the calculation: Total disintegrations = 37,000,000,000 × (60 × 60) = 37,000,000,000 × 3,600 = 133,200,000,000,000

Step 4: Write the final answer

The total number of disintegrations produced by one gram of radium in 1 hour is 133,200,000,000,000 disintegrations.

Key Concepts

Disintegrations per Second

Disintegrations per second is a unit commonly used in nuclear physics to describe radioactive decay. It measures how many nuclei within a sample decay every second.
Understanding this unit can help us interpret rates of radioactive processes and decay. For instance, if we know a sample's disintegration rate is 37 billion disintegrations per second, it suggests rapid decay, releasing particles or energy at a notably high rate.
In calculations, it is essential to correctly understand and utilize this unit, especially when transitioning into larger time frames, such as minutes, hours, or days.
When solving such problems, it helps to remember that larger durations naturally lead to much larger numbers of total disintegrations, considering the rapid rate inherent in these processes.

Conversion of Time Units

Time conversion is crucial for solving problems that involve rates over different periods. In this exercise, we need to convert hours to seconds to align our units with the disintegration rate stated in seconds.
Here's how to do it step by step:

• Recognize that 1 hour is equivalent to 60 minutes.
• Since each minute contains 60 seconds, multiply 60 by 60 to find the total seconds in an hour.
• This calculation gives us: 1 hour = 60 x 60 = 3,600 seconds.

Being meticulous with such conversions ensures that our calculations remain consistent and accurate throughout the problem.
Common time unit conversions include: hours to minutes, minutes to seconds, and vice versa. Mastering these conversions is invaluable for any problem involving time-based rates.

Multiplication in Scientific Notation

Scientific notation is a helpful tool for dealing with extremely large or small numbers, especially in scientific fields like physics. The format involves expressing numbers as a product of a number between 1 and 10 and a power of ten.
This problem involves multiplying using scientific notation, which simplifies large-number calculations considerably.
Here's how to approach it:

• Convert large numbers into scientific notation. For instance, 37 billion is written as \(3.7 \times 10^{10}\).
• Multiply this by the total seconds obtained from the time conversion. Here, \(3.6 \times 10^{3}\) seconds for an hour.
• When multiplying in scientific notation, multiply the leading numbers and add the exponents: \((3.7 \times 3.6) \times 10^{10+3}\).
• This simplifies to \(13.32 \times 10^{13}\), or \(1.332 \times 10^{14}\) in a more succinct form.

Using scientific notation not only makes calculations cleaner but also reduces the risk of error when handling cumbersome numbers and tedious multiplication.