Problem 37 - Applied Calculus

Question

A new DVD is available for sale in a store one week after its release. The cumulative revenue, \(\$ R\), from sales of the DVD in this store in week \(t\) after its release is $$ R=f(t)=350 \ln t \quad \text { with } t>1 $$ Find \(f(5), f^{\prime}(5)\), and the relative rate of change \(f^{\prime} / f\) at \(t=5 .\) Interpret your answers in terms of revenue.

Step-by-Step Solution

Answer

At week 5, the revenue is \$563.15, increasing at \$70/week, with a relative growth rate of 12.42% per week.

Step 1: Understand the problem

We need to find the cumulative revenue, the rate of change of revenue, and the relative rate of change at week 5.

Step 2: Calculate cumulative revenue, \( f(5) \)

We'll substitute \( t = 5 \) into the function \( f(t) = 350 \ln t \).Compute:\[ f(5) = 350 \ln(5) \approx 350 \times 1.609 = 563.15 \]Thus, the cumulative revenue at week 5 is approximately \$563.15.

Step 3: Differentiate the function, \( f'(t) \)

Differentiate the given function to find the rate of change:\[ f(t) = 350 \ln t \implies f'(t) = 350 \frac{1}{t} \]This represents the change in revenue with respect to time.

Step 4: Calculate the rate of change, \( f'(5) \)

Substitute \( t = 5 \) into \( f'(t) = 350 \frac{1}{t} \) to find:\[ f'(5) = 350 \frac{1}{5} = 70 \]This means revenue is increasing at a rate of \$70 per week in week 5.

Step 5: Find the relative rate of change, \( \frac{f^{\prime}(t)}{f(t)} \)

Calculate the relative rate of change at week 5 using the formula \( \frac{f'(t)}{f(t)} \):\[ \frac{f'(5)}{f(5)} = \frac{70}{563.15} \approx 0.1242 \]This relative rate shows the weekly percentage change.

Key Concepts

Cumulative Revenue

Cumulative revenue is the total income generated from sales up to a specific point in time. In the context of this exercise, we are looking at the sales of a DVD that are captured by the function \( f(t) = 350 \ln t \). This function tells us how much revenue has accumulated by week \( t \) after the DVD's release.
To find the cumulative revenue at week 5, we substitute \( t = 5 \) into the function. This gives us \( f(5) = 350 \ln(5) \), which calculates approximately to \\(563.15.
This means that by week 5, the store has made about \\)563.15 from selling the DVD. Understanding cumulative revenue is crucial as it provides a snapshot of total sales at any given point, which can help businesses plan and forecast for the future.

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point. In other words, it helps us understand how fast the revenue is increasing as time progresses.
For the revenue function \( f(t) = 350 \ln t \), differentiation gives us \( f'(t) = 350 \frac{1}{t} \). This derivative represents the rate of change of revenue concerning time. It's important because it shows how the sales trend might shift over short periods.
By calculating derivatives, businesses can strategize how to increase revenue based on current sales trends and how they are expected to change over time.

Rate of Change

The rate of change is a crucial concept in understanding how one quantity changes with respect to another. In this exercise, we're looking at how revenue changes over time.
After differentiating the function \( f(t) = 350 \ln t \), we found \( f'(t) = 350 \frac{1}{t} \), which signifies the rate of revenue increase. Upon substituting \( t = 5 \), we get \( f'(5) = 70 \), indicating the revenue is increasing by \$70 per week during the fifth week.
Understanding the rate of change is vital for short-term planning. It helps businesses adjust sales targets, budgets, and inventories according to their current performance.

Relative Rate of Change

The relative rate of change is a measure of how fast a quantity changes relative to the size of the quantity at that point in time. It provides insight into the percentage change per unit time, which is extremely useful in identifying growth trends.
In the context of this problem, the relative rate of change at week 5 is calculated by dividing the rate of change \( f'(5) \) by the cumulative revenue \( f(5) \). The calculation \( \frac{f'(5)}{f(5)} \approx 0.1242 \) indicates approximately a 12.42% increase per week at that time.
Businesses often use relative rates of change to make comparative analyses of their growth rates over different periods or in different segments.