Question

One hundred grams of radium are stored in a container. The amount \(R\) (in grams) of radium present after \(t\) years can be modeled by \(R=100 e^{-0.00043 t}\). After how many years will only 5 grams of radium be present?

Short answer

By calculating the above expression, we find that the time \(t\) approximately equals to 1503 years. Therefore, it will take about 1503 years for the 100 grams of radium to decay to 5 grams.

Step-by-step solution

Set up the equation

Given the exponential decay model \(R = 100e^{-0.00043t}\), we are asked to find \(t\) when \(R = 5\). First, setup the equation with \(R = 5\), so the equation to be solved will be:\(5 = 100e^{-0.00043t}\)

Simplify the equation

Next step is to isolate the exponential part of the equation \(e^{-0.00043t}\). Divide both sides of the equation by 100 to get:\(e^{-0.00043t} = 0.05\)

Solve for \(t\)

Taking the natural logarithm on both sides to solve for \(t\), we have:\(-0.00043t = ln(0.05)\)Then, to isolate \(t\), divide both sides by -0.00043 to find: \(t = ln(0.05)/-0.00043\)

Key concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. This base is a peculiar mathematical constant that appears in various aspects of calculus and complex numbers, especially in growth and decay processes. When we take the natural logarithm of a number, we're asking the question: "To what power must \( e \) be raised to get this number?"

For example, in our exercise related to radioactive decay, the equation \( -0.00043t = \ln(0.05) \) involves taking the natural logarithm of 0.05 to isolate \( t \). This is because the natural logarithm allows us to rewrite an equation involving an exponent of \( e \) into a more manageable linear form.

• Logarithms help solve equations involving exponential functions.
• They allow you to transform multiplicative processes into additive ones, which are easier to handle.
• Particularly, natural logs are useful for solving equations involving base \( e \), which is common in continuous growth or decay models.

Exponential Function

An exponential function is a mathematical expression in the form \( y = ab^x \), where \( b \) is a positive constant. If \( b \) equals \( e \), we have the natural exponential function \( y = ae^x \). Exponential functions adore model growth and decay processes.

In the context of the radium decay problem, the function \( R = 100e^{-0.00043t} \) represents an exponential decay model. Here \( 100 \) is the initial quantity of radium, and the term \( e^{-0.00043t} \) encapsulates the decay over time.

This multiplication with \( e \) raises the power \( -0.00043t \), which defines how the radium decreases over time, showing a rate of decay. Particularly,

• Exponential decay implies that the quantity decreases by a consistent percentage over equal intervals of time.
• As time \( t \) increases, the effect of the negative exponent leads the function towards zero.
• Such models are crucial in fields like physics and biology, helping to predict how a quantity diminishes over time.

Radioactive Decay

Radioactive decay is a natural process where unstable nuclei lose energy by emitting radiation. This activity transforms the original nucleus into a different element or a different state. It's characterized as a random process on an atomic level for the decay of a single atom, but a predictable process on a large scale of numbers.

The decay is typically modeled using exponential decay functions, as seen here with the radium example. The decay model \( R = 100e^{-0.00043t} \) illustrates how the quantity of a radioactive substance reduces over time.

• Radioactive decay follows a first-order kinetics, meaning the decay rate is proportional to the current amount of substance.
• It's important for applications such as radiometric dating or calculating dosage in medical treatments.
• It provides insights into the permanence and decay rates of isotopes in scientific studies.

These decay processes exemplify how natural systems tend to reach a level of stability over time, which is mathematically expressed as an exponential decay model.