The amount of radium-224 in a sample was measured at the...

The amount of radium-224 in a sample was measured at the indicated times. \begin{tabular}{c|cccccccc} time in days & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline $\mathrm{mg}$ & 100 & 82.7 & 68.3 & 56.5 & 46.7 & 38.6 & 31.9 & 26.4 \end{tabular} (a) Estimate how much radium will be left after 10 days. (b) If the sample is considered to be safe when the amount of radium is less than .01 mg, estimate how long the sample needs to be stored before it can be safely disposed of.

The amount of radium-224 in a sample was measured at the indicated times.
\begin{tabular}{c|cccccccc}
time in days & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline $\mathrm{mg}$ & 100 & 82.7 & 68.3 & 56.5 & 46.7 & 38.6 & 31.9 & 26.4
\end{tabular}
(a) Estimate how much radium will be left after 10 days.
(b) If the sample is considered to be safe when the amount of radium is less than .01 mg, estimate how long the sample needs to be stored before it can be safely disposed of.

Key Concepts

Exponential Decay

Exponential decay is a process in which a quantity decreases at a rate proportional to its current value, leading to a rapid early decrease that slows over time. In many natural and physical phenomena, including radioactive decay, the relationship is modeled using an exponential function, commonly written as y = y?e^(?kt). This concept underlies the analysis and prediction of the remaining quantity of a substance after a given period.

Decay Constant

The decay constant, represented by k in the exponential decay formula, quantifies the rate at which a substance decays. It is determined by using observed data points and is crucial for making predictions about the future amount of a decaying material. A higher decay constant indicates a faster rate of decay, and it is essential for determining the time required for a substance to reduce to a specified level.

Logarithmic Functions

Logarithmic functions are used to solve equations where the unknown variable appears in the exponent, such as in exponential decay models. They allow for the isolation of the time variable when equating the decayed amount to a threshold value. This process is fundamental in calculating the necessary time for a decaying substance to reach a safe or desired level.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. It is inherently random but statistically predictable using exponential decay models. Understanding radioactive decay is essential in fields such as nuclear physics, medicine, and environmental science, particularly for determining the lifespan and safe disposal times of radioactive materials.

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