Problem 32 - Calculus Early Transcendentals

Question

Polonium-2 10 is a radioactive element with a half-life of 140 days. Assume that 10 milligrams of the element are placed in a lead container and that \(y(t)\) is the number of milligrams present t days later. (a) Find an initial-value problem whose solution is \(y(t) .\) (b) Find a formula for \(y(t) .\) (c) How many milligrams will be present after 10 weeks? (c) How many milligrams will be present after 10 weeks? (d) How long will it take for \(70 \%\) of the original sample to decay?

Step-by-Step Solution

Answer

After 10 weeks, about 7.56 mg remains; 70% decays in about 336 days.

Step 1: Understand the Problem

We need to find an initial value problem representing the decay of Polonium-210, which has a half-life of 140 days, and solve it to find formulas for decay over time. Then we will use this formula to determine how much of the substance remains after specific time intervals.

Step 2: Write the Initial-value Problem

The rate of decay for a radioactive substance can be expressed using first-order differential equations. The equation is given by \( \frac{dy}{dt} = -ky \) where \( k \) is the decay constant. The initial condition is \( y(0) = 10 \), representing the initial amount. Thus the initial-value problem is: \( \frac{dy}{dt} = -ky \), \( y(0) = 10 \).

Step 3: Calculate the Decay Constant

The decay constant \( k \) is found using the half-life formula \( T_{1/2} = \frac{\ln(2)}{k} \). Given the half-life is 140 days, \( k = \frac{\ln(2)}{140} \).

Step 4: Solve the Differential Equation

Solving \( \frac{dy}{dt} = -ky \), we get \( y(t) = y(0) e^{-kt} \). Substituting the initial condition \( y(0) = 10 \), we find: \[ y(t) = 10 e^{-0.00495t} \].

Step 5: Calculate the Amount after 10 Weeks

Convert 10 weeks to days: 10 weeks × 7 days/week = 70 days. Substitute \( t = 70 \) into the formula: \[ y(70) = 10 e^{-0.00495 \times 70} \]. Calculate to find \( y(70) \approx 7.56 \text{ mg} \).

Step 6: Calculate Time for 70% Decay

To find out when 70% of the original sample decays, meaning 30% remains, solve for \( t \) when \( y(t) = 0.3 \times 10 = 3 \). Using \( y(t) = 10 e^{-0.00495t} \), set the equation: \( 3 = 10 e^{-0.00495t} \). Solve for \( t \): \( e^{-0.00495t} = 0.3 \), \( -0.00495t = \ln(0.3) \), \( t \approx 336 \text{ days} \).

Key Concepts

Half-life

The concept of half-life is crucial to understanding radioactive decay. It refers to the time required for half of the radioactive substance to decay. In simpler terms, if you start with a certain amount of a radioactive element, in one half-life, you will have half of it remaining.
For Polonium-210, specifically, the half-life is 140 days. This means that every 140 days, only half of the initial amount will remain. Understanding half-life helps us calculate how much of a substance remains after any given time.
When you know the half-life, you can determine the decay constant, which is used in equations predicting decay over time. This knowledge allows for precise calculations in various scientific and medical applications, such as dating ancient artifacts and medically tracing radioactive isotopes.

Differential Equations

Differential equations play a pivotal role in modeling radioactive decay. They express the rate of change of a quantity based on its current state. In the context of radioactive decay, the rate at which the substance decays is proportional to the amount currently present.
The general form of the differential equation used here is \( \frac{dy}{dt} = -ky \), where \( k \) is the decay constant. The negative sign indicates a decrease in the substance over time.
This equation helps in framing the problem and defining how the substance decreases. Solving this differential equation under specific initial conditions allows us to predict the amount of substance remaining at any future time point. Such equations are fundamental in fields ranging from physics to engineering, providing powerful tools for modeling natural phenomena.

Decay Constant

The decay constant is a crucial factor in calculating radioactive decay. It provides the rate at which a radioactive substance decays. The decay constant is denoted by \( k \) and is related to the half-life of the substance by the formula: \( T_{1/2} = \frac{\ln(2)}{k} \).
In this exercise, we calculated \( k \) for Polonium-210 based on its half-life of 140 days to be \( 0.00495 \) per day. This constant is used in the equation \( y(t) = y(0) e^{-kt} \) to model exponential decay.
The decay constant is essential in predicting how quickly a substance undergoes decay and helps in diverse applications such as nuclear medicine and radiation safety assessments. A deeper understanding of \( k \) enables more accurate predictions and planning concerning the behavior of radioactive substances.

Exponential Decay

Exponential decay is a type of decay where the quantity decreases at a rate proportional to its current value. This concept is mathematically represented using an exponential function. In radioactive decay, the formula \( y(t) = y(0) e^{-kt} \) models this behavior.
Here, \( y(t) \) represents the amount of substance remaining at time \( t \), \( y(0) \) is the initial quantity, and \( k \) is the decay constant. The exponential decay pattern ensures that while the quantity never truly reaches zero, it decreases rapidly initially and then tapers off more gradually.
This model helps in predicting both short-term changes and long-term trends of the behavior of a radioactive substance. Understanding this concept is vital for manipulating and anticipating the outcomes in fields such as pharmacology, environmental science, and physics.