Use the radioactive decay model. The half-life of cesium-137...

Use the radioactive decay model. 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 $m(t)=m_{0} e^{-r t}$ that models the mass remaining after $t$ years. c. How much of the sample will remain after 80 years? d. After how many years will only 2 g of the sample remain?

Use the radioactive decay model.
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 $m(t)=m_{0} e^{-r t}$ that models the mass remaining after $t$ years.
c. How much of the sample will remain after 80 years?
d. After how many years will only 2 g of the sample remain?

Key Concepts

Exponential Decay

Exponential decay describes processes in which a quantity decreases at a rate proportional to its current value. This concept forms the basis of mathematical models used to represent phenomena like radioactive decay, where the mass of a substance decreases over time following a specific decay pattern.

Half-Life

The half-life is the period required for a quantity undergoing decay to reduce to half its initial value. It is a key concept in radioactive decay because it directly links the time scale of decay to the remaining amount of the substance, allowing predictions of the material's behavior over extended periods.

Decay Constant

The decay constant is a parameter that quantifies the probability of decay per unit time in continuous exponential decay models. It is closely related to the half-life through the relation r = ln(2)/h, enabling conversion between discrete- and continuous-time formulations of radioactive decay.

Logarithmic Functions

Logarithmic functions are used in solving for variables such as time in exponential decay equations. They allow for the rearrangement and solution of equations where the quantity decreases multiplicatively over time, which is crucial when determining the time required for a substance to decay to a given amount.

Mathematical Modeling of Radioactive Decay

This involves setting up equations that accurately represent the decay process using exponential functions. Such models can use either a base-2 formulation, which directly incorporates the half-life, or a base-e formulation that uses the decay constant, providing versatile tools to analyze and predict the behavior of radioactive substances.

Transcript

00:01 Initial mass of the radioactive substance given is 10 gram that is consider m node as 10 gram and half life of the substance is 30 years.

00:13 Take h half life as 30 years.

00:16 Now in part a we need to model a function for our given decay model.

00:22 So mt equals m node into 2 raise 2 power minus t by h.

00:27 This is our standard decay model.

00:31 Put our value of m nod and h here so i'm not is 10 and h is 30 so this is our function which models are decay rate now in part b firstly we will find relative decay rate to express our given model to express our given function in exponential form therefore r is relative decay rate is l and 2 by h half so r becomes log 2 by h is 30 years.

01:10 So using calculator we can find this value which is 0 .0231 so our relative decay rate is 0 .0231 now writing this in exponential form so our function for decay model is mt equals 10 into e raised to power minus 0 .0231 into t.

01:35 Name this as equation number two.

01:37 Now in part c we need to find that what is the mass of substance left after 18 years? so put t equals 18 in equation number two.

01:50 So we get m t equals 10 into e raised to power minus 0 .0231 into t which is 18 after multiplication we get 10 into e raised to power minus 1 .848 sorry this was not 18 it's 80 years so we need to find our mass of substance after 80 years so we will put time as 80 and our and our mass after 80 years becomes 10 into e raised to power minus 1 .848 using calculator we can find this value which is 10 into 0 .1575 and after further multiplication we get our mass of substance remained after 80 years that is 1 .575 gram or we can approximate the mass of substance left after 80 years as 1 .6 gram...

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