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The rate at which a quantity of a radioactive substance decays is proportional to the quantity of the substance. The constant of proportionality λ is called the decay constant or decay rate (λ, like most physical constants, is conventionally represented by a positive number. That's why we call it a decay rate. If a quantity were increasing, we would use the words "growth rate" and also represent that by a positive number). Write the initial value problem that describes the amount N(t) of the radioactive substance remaining after time t, if the initial quantity N(0) is 6.

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anicalculus

Answer:

[tex]\frac{dN}{dt} =[/tex] λN           N(0) = 6

N(t) = N₀e^(λt)

Applying the inital value condition

N(t) = 6e^(λt)

Step-by-step explanation:

Summarizing  the information briefly and stating the  variables in the problem.

t = time elapsed during the decay

N(t) = the amount of the radioactive substance remaining after time t

λ= The constant of proportionality  is called the decay constant or decay rate

Given the initial conditions

N(0) = N₀ = 6

The rate at which a quantity of a radioactive substance decays ([tex]\frac{dN}{dt}[/tex]) is proportional to the quantity of the substance (N)  and λ is the constant of proportionality  is called the decay constant or decay rate :

[tex]\frac{dN}{dt} =[/tex] λN          

N(t) = N₀e^(λt) ......equ 1

substituting the value of  N₀ = 6 into equation 1

N(t) = 6e^(λt)

To solve this problem we must know about the differential equations.

The amount N(t) of the radioactive substance remaining after time t can be given as [tex]N(t)= 6\ e^{\lambda t}[/tex].

Given to us

  • The rate at which a quantity of a radioactive substance decays is proportional to the quantity of the substance.
  • The constant of proportionality λ is called the decay constant or decay rate
  • the initial quantity N(0) is 6.

Solution

According to the given information,

[tex]\dfrac{dN}{dt} =\lambda N[/tex]

[tex]\dfrac{dN}{N} =\lambda dt[/tex]

Integrating both the side of the equation we get,

[tex]\int\dfrac{dN}{N} =\int \lambda dt[/tex]

[tex]ln N=\lambda t+C[/tex]

Taking antilog,

[tex]N=e^{\lambda t+C}\\\\ N= e^{\lambda t}e^C\\\\ N = e^{\lambda t}N_o\\\\ N= N_o\ e^{\lambda t}[/tex]

Thus, the exponential function [tex]N(t)= N_o\ e^{\lambda t}[/tex] represents the amount of the radioactive substance remaining after time t.

where,

[tex]N_o[/tex] is the initial amount of radioactive substance,

[tex]\lambda[/tex] is the rate of decay or decay rate,

and t is the time period.

Substituting the amount of radioactive substance at the initial condition,

[tex]N(t)= N_o\ e^{\lambda t}\\ N(t)= 6\ e^{\lambda t}[/tex]

Hence, the amount N(t) of the radioactive substance remaining after time t can be given as [tex]N(t)= 6\ e^{\lambda t}[/tex].

Learn more about differential equations:

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