Question

Constant doubling time Prove that the doubling time for an exponentially increasing quantity is constant for all time.

Short answer

Question: Prove that the doubling time for an exponentially increasing quantity is constant for all time. Answer: The doubling time for an exponentially increasing quantity is given by the formula \(T_d = \frac{\ln{2}}{k}\), where \(k\) is the constant growth rate. This shows that the doubling time is constant and dependent only on the growth rate, and remains constant regardless of the value of the quantity or the time elapsed.

Step-by-step solution

Define Doubling Time

Doubling time is the amount of time it takes for a quantity to double in size or value. In the context of exponential growth, the doubling time remains constant throughout the entire growth process.

Model Exponential Growth

Exponential growth can be modeled by the equation: \( y(t) = y_0 e^{kt} \) where \(y(t)\) is the quantity at time \(t\), \(y_0\) is the initial quantity, \(k\) is the growth rate, and \(e\) is the base of the natural logarithm (\(\approx 2.718\)).

Identify the Doubling Time

Let the doubling time be denoted as \(T_d\). This is the time it takes for the quantity to double from its initial value, or when \(y(T_d) = 2y_0\). We can substitute this into the exponential growth equation and solve for \(T_d\): \( 2y_0 = y_0 e^{kT_d} \)

Solve for the Doubling Time

Divide both sides of the equation by \(y_0\): \( 2 = e^{kT_d} \) Next, we take the natural logarithm of both sides: \( \ln{2} = \ln{e^{kT_d}} \) Using the logarithmic identity \(\ln{a^b} = b\ln{a}\), we can simplify the right-hand side of the equation: \( \ln{2} = kT_d\ln{e} \) Since \(\ln{e} = 1\), we have: \( \ln{2} = kT_d \) Finally, we can solve for \(T_d\) by dividing both sides by \(k\): \(T_d = \frac{\ln{2}}{k}\)

Conclusion

We have found that the doubling time for an exponentially increasing quantity is given by the formula \(T_d = \frac{\ln{2}}{k}\). This expression shows that the doubling time is constant and dependent only on the growth rate \(k\). As long as the growth rate remains constant, so will the doubling time, regardless of the value of the quantity or the time elapsed. This proves that the doubling time for an exponentially increasing quantity is constant for all time.

Key concepts

Doubling Time Formula

Understanding the doubling time formula is essential to grasp how quickly something grows exponentially. It tells us how long it will take for a particular quantity to double in size or value. Imagine you're watching a magic beanstalk grow, and you're curious how fast it will reach twice its size. The doubling time formula gives you that insight.

In mathematical terms, the formula is expressed as
\[\begin{equation}T_d = \frac{\ln{2}}{k}\end{equation}\]
where \(T_d\) is the doubling time, \(\ln{2}\) is the natural logarithm of 2 (approximately 0.693), and \(k\) is the constant growth rate. This formula derives from the properties of exponential functions and shows that the doubling time is solely dependent on the growth rate. If the growth rate stays constant, so does the doubling time, irrespective of any other variables.

For students, this concept is particularly relevant when studying populations, investments, or any scenario where exponential growth occurs. Knowing the doubling time can lead to more informed predictions and decisions.

Exponential Growth Model

The exponential growth model is like a blueprint for understanding how quantities expand over time. It could describe anything from the spread of a virus to the way your savings account balloons with compound interest. The main characteristic of exponential growth is that the rate of increase becomes faster as the amount itself gets larger.

The model is represented by the equation
\[\begin{equation}y(t) = y_0 e^{kt}\end{equation}\]
where \(y(t)\) is the quantity after time \(t\), \(y_0\) is the initial quantity, \(k\) is the constant growth rate, and \(e\) is the base of the natural logarithm, an irrational number approximately equal to 2.718.

When visualizing this model on a graph, you'll see a J-shaped curve, illustrating that as time progresses, the quantity doesn't just increase—it accelerates. This concept is crucial for students across various disciplines, from biology to finance, enabling them to predict future outcomes based on current growth patterns.

Natural Logarithm

The natural logarithm is a unique creature in the mathematical zoo. It's a way to measure the time needed to reach a certain level of growth under continuous compounding, and it’s denoted as \(\ln x\), with the special number \(e\) (approximately 2.718) as the base. It's called 'natural' because of its organic appearance in various natural processes, like radioactive decay or population growth.

The natural log of a number answers the question: 'To what power do we raise \(e\) to get this number?' So, for \(\ln{e}\), the answer is 1 because \(e^1 = e\). And for our doubling time formula, \(\ln{2}\) pops up because we’re interested in knowing when our quantity reaches twice its initial size.

The natural logarithm has a few properties that make it play nicely with exponential functions, like the identity
\[\begin{equation}\ln{a^b} = b \ln{a}\end{equation}\]
which allows us to untangle equations involving exponential growth and solve for unknowns like the doubling time.