Problem 1 - Calculus and its applications

Question

Find the half-life for each situation. An element loses \(12 \%\) of its mass every year.

Step-by-Step Solution

Answer

The half-life is approximately 5.66 years.

Step 1: Understand the Problem

We need to find out how long it takes for a substance to reduce to half its initial mass, given a loss of 12% per year.

Step 2: Set Up the Exponential Decay Model

The mass of a substance decays exponentially if it decreases by a constant percentage each period. The model for exponential decay is given by \(m(t) = m_0 (1 - r)^t\), where \(m_0\) is the original mass, \(r\) is the decay rate, and \(t\) is time in years. In this case, \(r = 0.12\).

Step 3: Apply the Half-Life Condition

For the half-life, we want the mass to be half of the initial mass, i.e., \(m(t) = \frac{1}{2}m_0\). Substituting this in gives the equation: \(\frac{1}{2}m_0 = m_0 (1 - 0.12)^t\).

Step 4: Simplify the Equation

Divide both sides of the equation by \(m_0\) to eliminate \(m_0\): \(\frac{1}{2} = (0.88)^t\). Now we need to solve this equation for \(t\).

Step 5: Solve for t Using Logarithms

Take the natural logarithm of both sides to solve for \(t\). \(\ln \left( \frac{1}{2} \right) = t \ln(0.88)\).

Step 6: Isolate t

To find \(t\), divide both sides by \(\ln(0.88)\): \(t = \frac{\ln(0.5)}{\ln(0.88)}\).

Step 7: Calculate t

Use a calculator to find \(\ln(0.5)\) and \(\ln(0.88)\), then compute \(t\). This will give \(t \approx 5.66\) years.

Key Concepts

Half-Life

The half-life of a substance refers to the time it takes for half of the substance to decay or reduce in quantity. This concept is pivotal when discussing the decay of radioactive materials, as well as certain chemicals and biological substances. The half-life is a constant; it does not change over time and is independent of the initial amount of the substance. This means that whether you start with an ounce or a ton, the time it takes for the substance to reduce to half remains the same. In the context of exponential decay, understanding half-life helps us predict how substances diminish over consistent periods of time.

Exponential Decay Model

The exponential decay model is a mathematical representation that's used to describe processes where quantities decrease by a fixed percentage over time. This model is applicable in various domains like physics, chemistry, and finance. It is expressed as

• \(m(t) = m_0(1 - r)^t\)

Here:

• \(m(t)\) represents the mass after time \(t\)
• \(m_0\) is the initial mass
• \(r\) is the decay rate,

This formula models how a quantity diminishes over time. In most real-world scenarios, the decay rate \(r\) is given as a percentage. For instance, if a substance decays by 12% annually, \(r\) would be 0.12. As time progresses, the quantity approaches zero but never reaches it, highlighting the perpetual nature of exponential decay.

Natural Logarithm

A natural logarithm, denoted as \( \ln \), is a logarithm to the base of \(e\), where \(e\) is an irrational number approximately equal to 2.71828. Natural logarithms are crucial for solving equations in exponential decay, as they help unravel terms where the unknown variable is an exponent. When finding the half-life of a decaying substance, you often reach a point where you need to isolate the time variable \(t\) from an equation like: \((0.88)^t = \frac{1}{2}\) Taking the natural logarithm of both sides results in: \(\ln((0.88)^t) = \ln\left(\frac{1}{2}\right)\) Using logarithmic properties, you can then express this as: \( t \cdot \ln(0.88) = \ln\left(\frac{1}{2}\right)\) This step makes it possible to solve for \(t\) and is a key process in decay rate computation.

Decay Rate Calculation

Calculating the decay rate involves determining the constant percentage by which a substance decreases over a time period. This value is key in developing the exponential decay model. Given a problem where a substance loses, say 12% of its mass each year, the decay rate \(r\) is calculated directly from this percentage:

• \(r = 0.12\)

To find how many years it takes to reach half-life, you use the equation: \[\frac{1}{2} = (1 - r)^t\] Solving for \(t\) involves taking logarithms, as \(t\) is an exponent. Once the decay rate is known, you proceed with the equation to calculate the half-life by: \[t = \frac{\ln(0.5)}{\ln(1 - r)}\] In engaged contexts like this, having a precise decay rate impacts reliable predictions and analysis of how quickly something is reducing, which is essential across various disciplines.