Question

Using inverse relations One hundred grams of a particular radioactive substance decays according to the function \(m(t)=100 e^{-t / 650},\) where \(t>0\) measures time in years. When does the mass reach 50 grams?

Short answer

It will take approximately 450.76 years for the mass of the radioactive substance to reduce to 50 grams.

Step-by-step solution

Set up the equation m(t) = 50

In order to find the value of \(t\) when the mass is 50 grams, we need to set up the equation \(m(t) = 50\). The given mass decay function is \(m(t) = 100 e^{-t / 650}\), so the equation becomes: $$ 50 = 100 e^{-t / 650} $$

Solve for t

Now we need to solve for \(t\) in the equation \(50 = 100 e^{-t / 650}\). First, we will divide both sides of the equation by 100: $$ \frac{50}{100} = e^{-t / 650} $$ Simplify: $$ 0.5 = e^{-t / 650} $$ To solve for \(t\), we need to get rid of the exponential function. We can do this by taking the natural logarithm of both sides: $$ \ln(0.5) = \ln(e^{-t / 650}) $$ Now, apply the logarithm property \(\ln(a^b) = b \cdot \ln(a)\): $$ \ln(0.5) = -\frac{t}{650} \cdot \ln(e) $$ Since \(\ln(e) = 1\), the equation simplifies to: $$ \ln(0.5) = -\frac{t}{650} $$ Now, multiply both sides by -650: $$ -650 \cdot \ln(0.5) = t $$ Use a calculator to find the numerical value: $$ t \approx 450.76 $$ So, it takes approximately 450.76 years for the mass of the radioactive substance to reduce to 50 grams.

Key concepts

Exponential Decay

Exponential decay describes a process where the quantity of a substance decreases at a rate which is proportional to its current value. This concept is central in understanding radioactive decay. It means that as more mass is lost, the rate of decay slows down over time, but never fully stops.
The general form of an exponential decay function can be written as:

• \(m(t) = m_0 \, e^{-kt}\)

where:

• \(m(t)\) is the remaining quantity at time \(t\)
• \(m_0\) is the initial quantity
• \(k\) is the decay constant, which is characteristic of the material
• \(e\) is the base of the natural logarithm (approximately 2.718)

In our context with the cartoonish radioactive substance, it starts at 100 grams and decreases over time following this exponential pattern.

Half-Life

Half-life is a term used to describe the time taken for half of a radioactive substance to decay. In the context of the exercise, it helps understand how quickly the substance reduces. If you know the half-life of a substance, you can predict how much of it will remain after a given period.
For exponential decay, the half-life \(T_{1/2}\) can be calculated using the decay constant, \(k\), through the formula:

• \(T_{1/2} = \frac{\ln(2)}{k}\)

Given that we know our decay function is \(m(t) = 100 e^{-t / 650}\), the decay constant \(k\) is \(\frac{1}{650}\). Using this, the half-life can be easily calculated as follows:

• \(T_{1/2} = 650 \ln(2)\) years.

This result tells you how much time it takes for the mass to reduce from 100 grams to 50 grams.

Natural Logarithm

Natural logarithm, usually abbreviated as \(\ln\), is a mathematical function that is extremely handy when dealing with exponential growth or decay. In the exercise, the natural logarithm was used to solve for the time \(t\) when the mass becomes 50 grams.
The natural logarithm has the base \(e\), which is approximately 2.718. It's often used in equations involving exponential functions because it allows us to solve for the exponent. For example, if you want to find \(t\) in the equation \(m(t) = m_0 e^{-kt}\), you would:- Use \(\ln\) to remove the exponential base \(e\).- Apply the property \(\ln(a^b) = b \ln(a)\) to simplify your calculations.
This way, \(\ln\) transforms our exponential equation into a simpler linear form that is much easier to work with. The handy property of the natural logarithm, that \(\ln(e) = 1\), simplifies these equations further, facilitating easy calculations.