Answer
Verified
412.5k+ views
Hint:Here in this question we have to find the exponential decay rate. That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time t. So we can introduce a proportionality constant. Then further applying an integration on both sides and on simplification we get the required result.
Complete step by step answer:
Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be expressed by the formula \[y = a{\left( {1 - b} \right)^x}\] where y is the final amount, a is the original amount, b is the decay factor, and x is the amount of time that has passed. Exponential decays typically start with a differential equation of the form:
\[ \Rightarrow \,\,\dfrac{{dN}}{{dt}}\alpha \, - N(t)\]
That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time t. So we can introduce a proportionality constant \[\alpha \]:
\[ \Rightarrow \,\,\dfrac{{dN}}{{dt}} = - \alpha \,N(t)\]
We will now solve the equation to find a function of \[N(t)\]:
\[ \Rightarrow \,\,\dfrac{{dN}}{{N(t)}} = - \alpha \,dt\]
\[ \Rightarrow \,\,\dfrac{{dN}}{N} = - \alpha \,dt\]
Apply integration both sides, then
\[ \Rightarrow \,\,\int {\dfrac{{dN}}{N}} = \int { - \alpha } \,dt\]
\[ \Rightarrow \,\,\int {\dfrac{{dN}}{N}} = - \alpha \int {dt} \,\]
Using the integration formula \[\int {\dfrac{1}{x}dx = \ln x + c} \] and \[\int {dx = x + c} \], where c is an integrating constant.
\[ \Rightarrow \,\,\ln N = - \alpha t + c\,\]
As we know the logarithm function is the inverse form of exponential function, then
\[ \Rightarrow \,\,N = {e^{ - \alpha \,\,t + c}}\,\]
Or it can be written as:
\[ \therefore \,\,N = A{e^{ - \alpha \,t}}\]
Where A is a constant.
Hence, the general form of exponential decay rate is \[N = A{e^{ - \alpha \,t}}\].
Note:Exponential decay describes the process of reducing an amount by a constant percentage rate over a period of time. The integration is inverse of the differentiation so to cancel differentiation the integration is applied. Likewise the logarithm is inverse of exponential. Hence by using these concepts we obtain answers.
Complete step by step answer:
Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be expressed by the formula \[y = a{\left( {1 - b} \right)^x}\] where y is the final amount, a is the original amount, b is the decay factor, and x is the amount of time that has passed. Exponential decays typically start with a differential equation of the form:
\[ \Rightarrow \,\,\dfrac{{dN}}{{dt}}\alpha \, - N(t)\]
That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time t. So we can introduce a proportionality constant \[\alpha \]:
\[ \Rightarrow \,\,\dfrac{{dN}}{{dt}} = - \alpha \,N(t)\]
We will now solve the equation to find a function of \[N(t)\]:
\[ \Rightarrow \,\,\dfrac{{dN}}{{N(t)}} = - \alpha \,dt\]
\[ \Rightarrow \,\,\dfrac{{dN}}{N} = - \alpha \,dt\]
Apply integration both sides, then
\[ \Rightarrow \,\,\int {\dfrac{{dN}}{N}} = \int { - \alpha } \,dt\]
\[ \Rightarrow \,\,\int {\dfrac{{dN}}{N}} = - \alpha \int {dt} \,\]
Using the integration formula \[\int {\dfrac{1}{x}dx = \ln x + c} \] and \[\int {dx = x + c} \], where c is an integrating constant.
\[ \Rightarrow \,\,\ln N = - \alpha t + c\,\]
As we know the logarithm function is the inverse form of exponential function, then
\[ \Rightarrow \,\,N = {e^{ - \alpha \,\,t + c}}\,\]
Or it can be written as:
\[ \therefore \,\,N = A{e^{ - \alpha \,t}}\]
Where A is a constant.
Hence, the general form of exponential decay rate is \[N = A{e^{ - \alpha \,t}}\].
Note:Exponential decay describes the process of reducing an amount by a constant percentage rate over a period of time. The integration is inverse of the differentiation so to cancel differentiation the integration is applied. Likewise the logarithm is inverse of exponential. Hence by using these concepts we obtain answers.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE