Problem 38 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

SUBSCRIPTION GROWTH The publishers of a national magazine have found that the fraction of subscribers who remain subscribers for at least \(t\) years is \(f(t)=e^{-t / 10}\). Currently, the magazine has 20,000 subscribers and it is estimated that new subscriptions will be sold at the rate of 1,000 per year. Approximately how many subscribers will the magazine have in the long run?

Step-by-Step Solution

Answer

The magazine will approximately have 10,000 subscribers in the long run.

Step 1: Understand the Fraction Function

Identify the function that describes the fraction of subscribers remaining after time t: \(f(t) = e^{-t / 10}\).

Step 2: Current Subscribers

Note the current number of subscribers which is 20,000.

Step 3: Rate of New Subscriptions

Recognize that new subscriptions are added at the rate of 1,000 per year.

Step 4: Finding the Long-Run Subscribers

To find the steady-state number of subscriptions, use the formula \(N = \frac{R}{|f'(0)|}\), where R is the rate of new subscriptions and \(f(t) = e^{-t / 10}\).

Step 5: Calculate the Derivative

Calculate the derivative of the given function at t = 0. \(f'(t) = -\frac{1}{10} e^{-t / 10}\). At t = 0, \(f'(0) = -\frac{1}{10}\).

Step 6: Calculate the Steady-State Subscriptions

Substitute the values into the formula: \(N = \frac{R}{|f'(0)|} = \frac{1000}{1/10} = 1000 \times 10 = 10,000\).

Key Concepts

exponential decay function

An exponential decay function describes a process where a quantity decreases at a constant rate over time. In this exercise, the fraction of subscribers who stay subscribed for at least a certain number of years is modeled by the function: \(f(t) = e^{-t / 10}\). This formula shows how the number of subscribers diminishes over time. This is essential in predicting how many subscribers will remain after each year.
Exponential decay functions are common in modeling real-world processes like population decline, radioactive decay, and, as in this case, the proportion of magazine subscriptions that continue over time.
Key points to remember:

• Exponential decay involves a continuous decrease.
• The base of the exponential function is the mathematical constant \(e\), which approximately equals 2.71828.
• In the function \(f(t) = e^{-t / 10}\), the fraction decreases as time \(t\) increases.

steady-state subscribers

Steady-state subscribers refer to the number of subscribers that stabilizes once the system reaches equilibrium. In the context of the exercise, it means the number of subscribers the magazine ends up with in the long run.
To find the steady-state, we use the formula \(N = \frac{R}{|f'(0)|}\) where:

• \(N\) represents the steady-state number of subscribers.
• \(R\) is the rate at which new subscriptions are added, which is 1,000 per year.
• \(f'(0)\) is the derivative of the fraction function evaluated at \(t = 0\).
• The derivative of the function \(f(t) = e^{-t / 10}\) is \(f'(t) = -\frac{1}{10} e^{-t / 10}\).
• At \(t = 0\), \(f'(0) = -\frac{1}{10}\).

Substituting these values into the formula, we have: \(N = \frac{1000}{1/10} = 1000 \times 10 = 10,000\).
This shows that in the long run, the magazine will have 10,000 subscribers.

rate of change in subscriptions

The rate of change in subscriptions indicates how fast the number of subscribers is changing at any point in time. It's derived from the function that models subscriber retention. In this case, the function is \(f(t) = e^{-t / 10}\).
The derivative of this function, \(f'(t) = -\frac{1}{10} e^{-t / 10}\), helps to understand the rate at which subscriptions decay over time.

• A negative derivative, \(f'(0) = -\frac{1}{10}\), shows that the number of subscribers is decreasing.
• The magnitude of the derivative indicates how fast the decay occurs.
• Since the decay factor is \(-\frac{1}{10}\), it tells us that for each increment in time, the fraction of subscribers decreases by \(\frac{1}{10}\)th of the original rate.

Understanding this rate is crucial for predicting future trends and making business decisions related to sustaining magazine subscriptions.