17-24 . These exercises use the radioactive decay model....

$17-24$ . These exercises use the radioactive decay model. Radioactive Radium The half-life of radium- 226 is 1600 years. Suppose we have a 22 -mg sample. (a) Find a function $m(t)=m_{0} 2^{-4 / h}$ that models the mass remaining after $t$ years. (b) Find a function $m(t)=m_{0} e^{-r t}$ that models the mass remaining after $t$ years. (c) How much of the sample will remain after 4000 years? (d) After how long will only 18 mg of the sample remain?

$17-24$ . These exercises use the radioactive decay model.
Radioactive Radium The half-life of radium- 226 is 1600 years. Suppose we have a 22 -mg sample.
(a) Find a function $m(t)=m_{0} 2^{-4 / h}$ that models the mass remaining after $t$ years.
(b) Find a function $m(t)=m_{0} e^{-r t}$ that models the mass remaining after $t$ years.
(c) How much of the sample will remain after 4000 years?
(d) After how long will only 18 mg of the sample remain?

Key Concepts

Exponential Decay

Exponential decay describes a process where a quantity diminishes at a rate proportional to its current value. It is commonly used to model phenomena like radioactive decay, where the initial amount reduces by a constant fraction over equal time periods.

Half-Life

The half-life is the time required for a substance to reduce to half its initial amount. In radioactive decay processes, it serves as a key parameter in determining how quickly a material decreases and is incorporated directly into the decay models.

Exponential Decay Function (Base 2 Model)

This model uses a base of 2 in its exponential expression to reflect the property of half-life directly, where the quantity is halved after each half-life period. It is typically written in the form m(t) = m0 · 2^(-t/T) when T is the half-life.

Exponential Decay Function (Natural Exponential Model)

This form of the decay function employs the natural exponential base e, and is written as m(t) = m0 · e^(-rt). The parameter r (decay constant) is related to the half-life and provides an equivalent formulation of the decay process in continuous time.

Conversion Between Exponential Bases

Understanding how to convert between the base 2 model and the natural exponential model is important. This involves relating the decay constant r to the half-life by means of logarithmic transformations, ensuring that both models describe the same physical process.

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