Problem 75 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

REAL ESTATE INVENTORY In a certain community the fraction of the homes placed on the market that remain unsold for at least \(t\) weeks is approximately \(f(t)=e^{-0.2 t}\). If 200 homes are currently on the market and if additional homes are placed on the market at the rate of 8 per week, approximately how many homes will be on the market 10 weeks from now?

Step-by-Step Solution

Answer

Approximately 64 homes will be on the market after 10 weeks.

Step 1: Understand the Function

The fraction of homes remaining unsold after time t weeks is given by the function \[ f(t) = e^{-0.2t} \].

Step 2: Calculate Initial Unsold Homes

Initially, 200 homes are on the market. After 10 weeks, the fraction of these that remain unsold is \[ f(10) = e^{-0.2 \times 10} = e^{-2} \]. So, the number of unsold homes out of the initial 200 is \[ 200 \times e^{-2} \].

Step 3: Calculate Additional Homes Placed on Market

Additional homes are being placed on the market at a rate of 8 homes per week. Over 10 weeks, the total number of additional homes is \[ 8 \times 10 = 80 \].

Step 4: Calculate Unsold Additional Homes

Each week, additional homes placed on the market will have different periods to remain on the market. For instance, homes placed on the 1st week will remain for 9 weeks, 2nd week for 8 weeks, etc. Summing them up, the unsold fractions are calculated as: \[ \text{Total unsold additional homes} = 8 \times \bigg( \frac{1-e^{-10\times 0.2}}{e^{-0.2}} \bigg) \].

Step 5: Sum Total Unsold Homes

Add the unsold homes from the initial 200 homes and the unsold additional homes to get the total number of homes on the market after 10 weeks: \[ \text{Total unsold homes} = 200 \times e^{-2} + 8 \times \frac{1-e^{-2}}{0.2} \].

Step 6: Simplify and Solve

After calculating the values we get: \[ 200 \times e^{-2} + 40 \times (1 - e^{-2}) \]. Approximate the result using values of \[ e^{-2} \approx 0.135 \text{giving us} 27 + 37.3 \approx 64.3 \].

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept where a quantity decreases at a rate proportional to its current value. In the context of real estate, the fraction of unsold homes over time can be modeled using an exponential decay function. This is represented by the formula:
\[ f(t) = e^{-0.2t} \] Here, \( e \) is the base of the natural logarithm, and the exponent \( -0.2t \) represents the decay rate. The negative sign indicates a decrease and the coefficient 0.2 controls how fast the decay happens.

When you want to find the fraction of homes that remain unsold after a specific time \( t \), you substitute \( t \) into the function. In our exercise, when \( t \) is 10 weeks, the fraction of homes that are still unsold is calculated as:
\[ f(10) = e^{-0.2 \times 10} = e^{-2} \] By understanding this principle, students can recognize how exponential decay affects various real-life situations outside of real estate, such as radioactive decay in physics or depreciation of assets in economics.

Calculus in Economics

Calculus plays a significant role in economics, aiding in analysis, optimization, and understanding of dynamic systems. In our real estate example, calculus helps us determine how many homes remain unsold over time.

Using derivatives allows us to find rates of change, which is key in economics. For instance, understanding how the rate at which homes are placed on the market influences the overall inventory is crucial. The derivative of an exponential function helps us understand how quickly the number of unsold homes decreases.

In economic models, we often encounter functions that describe supply and demand over time. Calculus helps us to optimize these models by finding maximum or minimum values using derivatives, thereby informing better decision-making. Additionally, integrals can accumulate quantities over a period, much like summing the unsold homes in different weeks, as seen in our exercise. Overall, calculus provides the tools to model and solve real-world economic problems efficiently.

Integrals in Real-Life Applications

Integrals are a fundamental concept in calculus used to find the total accumulation of quantities. In the real-world scenario of the exercise, we use integrals to sum the total number of unsold additional homes over 10 weeks.

When homes are introduced to the market each week, each tranche of homes will have a different duration of time until the 10-week mark. The formula to calculate the total unsold additional homes incorporates the summation of these decaying fractions:
\[ \text{Total unsold additional homes} = 8 \times \bigg( \frac{1-e^{-10\times 0.2}}{e^{-0.2}} \bigg) \]
Integrals extend beyond real estate, applicable in fields like physics for calculating areas under curves, or in biology for estimating population growth over time. They help solve problems where straightforward summation of parts is required. By understanding and applying integrals, students gain a powerful tool for compiling and interpreting data in various disciplines.