Question

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.

Answer 1

Answer:

$\frac{dN}{dt} =$ 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 ($\frac{dN}{dt}$) is proportional to the quantity of the substance (N)  and λ is the constant of proportionality  is called the decay constant or decay rate :

$\frac{dN}{dt} =$ λN          

N(t) = N₀e^(λt) ......equ 1

substituting the value of  N₀ = 6 into equation 1

N(t) = 6e^(λt)