Find k such that p=p0 e^k t has the given doubling time. 10...

Name: Problem 33, Chapter 1: Calculus Single Variable
Uploaded: 2021-04-28T16:38:22-08:00
Duration: 1 min 17 s
Channel: Prashansha Kaushik
Description: Find k such that p=p0 e^k t has the given doubling time. 10

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Key Concepts

Logarithms in Exponential Equations

Logarithms are crucial for solving equations involving exponentials. Taking the natural logarithm on both sides of an equation like 2 = e^(kT) allows one to isolate the rate constant k. This operation leverages the property that the natural logarithm is the inverse of the exponential function, leading to the formula k = ln(2)/T in the case of doubling time.

Exponential Growth Models

Exponential growth models describe processes where the rate of change of a quantity is proportional to the current amount of the quantity. The general form of the equation, p = p?e^(kt), captures situations such as population growth and radioactive decay, with the constant k representing the rate of growth or decay.

Doubling Time

Doubling time is the period required for a quantity undergoing exponential growth to double its initial value. It is a useful measure in many applications such as biology, finance, and physics, providing an intuitive understanding of the speed of growth. In the context of the exponential growth model, setting the popular equation equal to twice the initial value leads to the relation 2 = e^(kT), where T is the doubling time.

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