Problem 42 - Applied Calculus

Question

During a recession a firm's revenue declines continuously so that the revenue, \(R\) (measured in millions of dollars), in \(t\) years' time is given by \(R=5 e^{-0.15 t}\) (a) Calculate the current revenue and the revenue in two years' time. (b) After how many years will the revenue decline to \(\$ 2.7\) million?

Step-by-Step Solution

Answer

(a) Current revenue: $5M; in 2 years: $3.70M. (b) Revenue declines to $2.7M in ≈4.11 years.

Step 1: Understanding the Formula

The revenue equation given is \(R = 5e^{-0.15t}\), where \(R\) is the revenue in millions of dollars and \(t\) is the time in years. The equation describes how revenue decreases over time as an exponential decay function.

Step 2: Calculate Current Revenue (t=0)

To find the current revenue, set \(t = 0\) in the equation: \[R = 5e^{-0.15(0)} = 5e^{0} = 5 \]Thus, the current revenue is $5 million.

Step 3: Calculate Revenue after 2 Years (t=2)

Substitute \(t = 2\) into the equation: \[R = 5e^{-0.15 imes 2} = 5e^{-0.3}\]Using a calculator, approximate \(e^{-0.3} \approx 0.7408182\), then:\[R \approx 5 \times 0.7408182 = 3.704091 \]Thus, the revenue in two years is approximately $3.70 million.

Step 4: Determine Time for Revenue to Decline to $2.7 million

Set \(R = 2.7\) and solve for \(t\) in the equation:\[2.7 = 5e^{-0.15t}\]First, divide both sides by 5:\[e^{-0.15t} = \frac{2.7}{5} = 0.54\]Take the natural logarithm of both sides:\[-0.15t = \ln(0.54)\]Solve for \(t\):\[t = \frac{\ln(0.54)}{-0.15} \approx \frac{-0.616186}{-0.15} \approx 4.107\]Thus, it will take approximately 4.11 years for the revenue to decline to $2.7 million.

Key Concepts

Revenue Forecasting

Revenue forecasting is a method used to estimate the future revenue of a company. This is crucial, especially during uncertain periods like a recession. Understanding how revenue might decrease helps businesses prepare and strategize effectively. In this exercise, the revenue is a continuously decreasing function of time due to economic downturns. By forecasting revenue, businesses can plan ahead to mitigate adverse financial impacts. Utilizing mathematical models, like exponential decay, can provide more accurate predictions of future revenue, ensuring the business remains financially stable.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In finance, these functions often model growth or decay processes. Within the exercise, the function \( R = 5e^{-0.15t} \) describes how the firm's revenue declines over time. In this case, the base of the exponential, \( e \), represents a natural constant approximated at 2.718. Meanwhile, the exponent \(-0.15t\) illustrates a decay rate. The negative sign indicates that the revenue decreases as time progresses. Exponential decay is commonly used to represent various real-world scenarios, such as radioactive decay, cooling processes, and depreciation of assets. Recognizing the nature of an exponential function helps in predicting how the revenue trends over time.

Natural Logarithms

Natural logarithms, denoted as \( \ln \), are the inverse operations to exponentiation with the base \( e \). They are crucial for solving equations involving exponential decays or growths, as seen in our revenue example. To find when the revenue will decline to a certain value, such as $2.7 million, natural logarithms are indispensable. By applying \( \ln \) to both sides of the equation \( e^{-0.15t} = 0.54 \), we obtain \(-0.15t = \ln(0.54)\). This process allows us to isolate the variable \( t \), making it possible to solve for time duration accurately. Understanding natural logarithms can simplify complex exponential problems, giving clear insights into the relationships within the function.