Problem 40 - Applied Calculus

Question

At the site of a spill of radioactive iodine, radiation levels were four times the maximum acceptable limit, so an evacuation was ordered. If \(R_{0}\) is the initial radiation level (at \(t=0\) ) and \(t\) is the time in hours, the radiation level \(R(t)\), in millirems/hour, is given by $$ R(t)=R_{0}(0.996)^{t} $$ (a) How long does it take for the site to reach the acceptable level of radiation of \(0.6\) millirems/hour? (b) How much total radiation (in millirems) has been emitted by that time?

Step-by-Step Solution

Answer

(a) Approximately 347.43 hours. (b) Approximately 691.42 millirems emitted.

Step 1: Establish Conditions for Acceptable Radiation Level

Initially, radiation levels are four times the acceptable limit. If the acceptable level is 0.6 millirems/hour, then initially, the radiation level \( R_0 = 4 \times 0.6 = 2.4 \) millirems/hour. We need to determine when \( R(t) = 0.6 \).

Step 2: Determine Time for Acceptable Level

Use the equation \( R(t) = R_0(0.996)^t \) and substitute \( R_0 = 2.4 \) and \( R(t) = 0.6 \):\[0.6 = 2.4(0.996)^t\]Divide both sides by 2.4:\[(0.996)^t = \frac{0.6}{2.4}\]\[(0.996)^t = 0.25\]Take the logarithm of both sides:\[\log((0.996)^t) = \log(0.25)\]Use the property of logarithms: \( t \cdot \log(0.996) = \log(0.25) \)Finally, solve for \( t \):\[t = \frac{\log(0.25)}{\log(0.996)}\]Calculate \( t \) using a calculator to find approximately 347.43 hours.

Step 3: Calculate the Total Radiation Emitted

The total radiation emitted up to time \( t \) can be found by integrating \( R(t) \) from 0 to \( t \):\[\text{Total Radiation} = \int_{0}^{t} R_0 (0.996)^t \, dt\]\[= R_0 \cdot \int_{0}^{t} (0.996)^t \, dt\]The integral \( \int (a^t) \, dt \) is \( \frac{a^t}{\ln(a)} + C \). Applying this:\[\int_{0}^{347.43} (0.996)^t \, dt = \frac{(0.996)^t}{\ln(0.996)} \bigg|_{0}^{347.43}\]\[= \frac{(0.996)^{347.43} - (0.996)^0}{\ln(0.996)}\]\[= \frac{0.25 - 1}{\ln(0.996)}\]Calculate the total radiation emitted, given \( R_0 = 2.4 \):\[\text{Total Radiation} = 2.4 \cdot \frac{-0.75}{\ln(0.996)}\]Use a calculator to evaluate this expression, resulting in approximately 691.42 millirems.

Key Concepts

Understanding Radiation Levels

Radiation level is a measure of the intensity of radiation in a given area. This is crucial, especially when dealing with radioactive substances, as high radiation levels can be harmful to health. At the accident site in the exercise, the initial radiation level is four times the maximum safety limit. This means that swift action is necessary to reduce exposure.

• The radiation level is measured in millirems per hour, where millirem is a unit of radiation dose.
• The site initially has a radiation level of 2.4 millirems/hour, which needs to be reduced to an acceptable level of 0.6 millirems/hour for safety.

To achieve this, we rely on the formula given: \[ R(t) = R_0(0.996)^t \]This equation shows how the radiation level decreases over time, considering the factor 0.996, which represents the decay constant. Understanding how radiation level changes over time is essential to determine when it is safe to re-enter a contaminated area. This leads us to explore the concept of half-life and logarithmic calculations to predict when the radiation will fall to safe levels.

Half-Life's Role in Radioactive Decay

Half-life is the time it takes for half of a radioactive substance to decay. It is a key concept in understanding radioactive decay and determining how long radioactive materials remain hazardous. In the context of the given problem, half-life helps us comprehend how quickly the radiation level will decrease over time.

• Each radioactive element has a unique half-life, which is critical for predicting its decay rate.
• The shorter the half-life, the faster the substance decays and becomes less dangerous. Longer half-lives mean persistent radiation levels.

Although the problem doesn't explicitly ask for the half-life, it's a good connection to the decay formula. Radioactive decay follows an exponential decay model, with the formula parameter 0.996 showing the fraction of radiation remaining over time. While not exactly providing the half-life value here, this decay model indicates the time it will take for any initial radiation level to diminish to a safer value.

Utilizing Logarithmic Equations

Logarithmic equations are mathematical tools that simplify the complex calculations involved in exponential decay processes like radioactive decay. In the exercise, the logarithm allows us to solve for time, determining when the radiation level reaches a safe limit.

• The formula \( R(t) = R_0(0.996)^t \) expresses the decay process, and to solve for \( t \), logarithms are essential since they help isolate the variable \( t \).
• When equations are in the form \( a^b = c \), taking the logarithm of both sides can help solve for unknown variables, particularly \( b \).

The use of logarithms involves these key steps:1. Take the logarithm of both sides of the equation: \[ \log[(0.996)^t] = \log(0.25) \]2. Use the logarithm property, \( t \cdot \log(0.996) = \log(0.25) \), to solve for \( t \). 3. Finally, compute \( t \) using a calculator, which results in approximately 347.43 hours.Understanding how to apply logarithmic equations helps demystify how long it will take for a radioactive site to return to safe radiation levels, which is key to planning safe evacuation and re-entry strategies.