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

$17-24$ . These exercises use the radioactive decay model. Radioactive Cesium The half-life of cesium- 137 is 30 years. Suppose we have a 10 -g sample. (a) Find a function $m(t)=m_{0} 2^{-t / h}$ that models the mass remaining after $t$ years. (b) Find a function $m(t)=m_{0} e^{-H}$ that models the mass remaining after $t$ years. (c) How much of the sample will remain after 80 years? (d) After how long will only 2 $\mathrm{g}$ of the sample remain?

$17-24$ . These exercises use the radioactive decay model.
Radioactive Cesium The half-life of cesium- 137 is 30 years. Suppose we have a 10 -g sample.
(a) Find a function $m(t)=m_{0} 2^{-t / h}$ that models the mass remaining after $t$ years.
(b) Find a function $m(t)=m_{0} e^{-H}$ that models the mass remaining after $t$ years.
(c) How much of the sample will remain after 80 years?
(d) After how long will only 2 $\mathrm{g}$ of the sample remain?

Key Concepts

Exponential Decay Model

The exponential decay model describes how a quantity decreases at a rate proportional to its current value. It is typically represented by equations of the form m(t) = m0 * a^(t) or m(t) = m0 * e^(-kt), where m0 is the initial amount, t is time, a is the decay factor per time unit, and k is the decay constant. This model is widely used in natural processes like radioactive decay and represents a continuous decrease over time.

Half-Life

The half-life of a substance is the time it takes for half of the substance to decay. In the context of exponential decay, the half-life governs the rate at which the amount decreases. The concept is essential for understanding processes like radioactive decay, where it provides a straightforward measure to determine how rapidly a substance loses its activity.

Decay Constant

The decay constant is a parameter in the exponential decay equation that quantifies the rate of decay. It is related to the half-life by the natural logarithm of 2, with the relationship k = ln(2) / (half-life). This constant plays a key role in converting between different exponential formulations and enables accurate predictions of the remaining quantity over time.

Logarithmic Equations

Logarithmic equations are often used to solve for the time or decay constant when given the remaining fraction of a substance. They are essential for working with exponential decay models, especially when determining how long it takes for a substance to reach a particular amount. By applying logarithms to both sides of an exponential equation, one can isolate the variable representing time or the decay constant.

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