Problem 37 - Algebra A Combined Function

Question

Objective C) Solve. Unless otherwise indicated, round results to one decimal place. See Example 7.One type of uranium has a radioactive decay rate of \(0.4 \%\) per day. If 30 pounds of this uranium is available today, how much will still remain after 50 days? Use \(y=30(0.996)^{x},\) and let \(x\) be 50

Step-by-Step Solution

Answer

After 50 days, approximately 23.3 pounds of uranium will remain.

Step 1: Identify the values

The given problem states that we are dealing with a radioactive decay rate of 0.4% per day. At the start, 30 pounds of uranium is available, and we want to find out how much will remain after 50 days. The formula provided is \(y = 30(0.996)^{x}\) where \(x\) represents the number of days, in this case, 50.

Step 2: Substitute the value of x

Insert the value of \(x = 50\) into the equation. This gives us \(y = 30(0.996)^{50}\).

Step 3: Calculate the power of the decay factor

Calculate \((0.996)^{50}\) using a calculator. This will give the decay factor after 50 days.

Step 4: Multiply by initial amount

Once you have the decay factor from Step 3, multiply it by the initial amount, which is 30 pounds. This gives the final amount remaining: \(y = 30 imes (0.996)^{50}\).

Step 5: Round the result

After calculating the multiplication from Step 4, round the result to one decimal place as instructed in the problem to find the remaining amount of uranium.

Key Concepts

Exponential Decay Formula

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. This is common in phenomena such as radioactive decay, where a substance loses mass over time. The exponential decay formula can be expressed as:
\[ y = y_0 \times (1 - r)^x \]
where:

• \( y \) is the final amount remaining after time has elapsed.
• \( y_0 \) is the initial quantity, which in our case, is 30 pounds of uranium.
• \( r \) is the decay rate expressed as a decimal. Here, the decay rate is 0.4% which equals 0.004.
• \( x \) is the time period over which decay occurs, in days.

Using the formula, you can calculate how much of a radioactive substance will remain after a specific period. This formula reveals why only a small part of the uranium remains after so many days, illustrating how the decay factor, even if seemingly insignificant at 0.996, can lead to a significant reduction over time.

Uranium Decay

Uranium is a naturally radioactive element and undergoes decay over time. This decay process is a nuclear reaction where the uranium atom splits, releasing energy and reducing in mass. When uranium decays, it sheds some of its particles, transforming gradually into other elements.
One of the parameters that tell us how fast this transformation happens is the decay rate. In our example, it's given as 0.4% per day. Though this may appear slow, uranium's transformation across days is significant due to the compounding effect inherent in exponential calculations.
This exercise uses uranium to demonstrate how radioactive materials diminish, shedding light on why managing and disposing of radioactive substances is crucial in maintaining safety and environmental standards.

Rounding Decimal Places

Rounding decimals is a mathematical process used to denote numbers in a simpler form with fewer decimal places. In many calculations, especially in scientific contexts, the results can have numerous decimals, and rounding helps in presenting these numbers more manageably.
To round to one decimal place, you look at the second decimal digit:

• If it's 5 or more, round up the first decimal digit by one.
• If it's less than 5, keep the first decimal digit as is.

In the final step of the uranium decay calculation, after determining \( y \), it is essential to round the result to one decimal place for clarity and adherence to the exercise requirements. This simple rounding principle helps maintain precision and consistency across scientific calculations.