Problem 73 - Applied Calculus

Question

A real estate office is selling condominiums at the rate of \(100 e^{-x / 4}\) per week after \(x\) weeks. How many condominiums will be sold during the first 8 weeks?

Step-by-Step Solution

Answer

Approximately 347 condominiums will be sold in the first 8 weeks.

Step 1: Understanding the Problem

The given function for the number of condominiums sold per week is \(100 e^{-x / 4}\). We need to find the total number of condominiums sold over the first 8 weeks. This requires finding the definite integral of this function from \(x = 0\) to \(x = 8\).

Step 2: Set Up the Integral

We need to calculate the definite integral of the function. The integral to calculate is: \[\int_0^8 100 e^{-x / 4} \, dx\]

Step 3: Compute the Antiderivative

The antiderivative of \(100 e^{-x / 4}\) can be found using the substitution method. Set \(u = -x / 4\), then \(du = -1/4 \, dx\) or \(dx = -4 \, du\). This gives us the integral: \(-400 \int e^{u} \, du\). The antiderivative then becomes \(-400 e^u + C\).

Step 4: Back-Substitute and Evaluate the Definite Integral

Replace \(u\) with \(-x/4\) to get the original variable back into our antiderivative: \(-400 e^{-x/4}\). We now compute \[\int_0^8 100 e^{-x / 4} \, dx = \left[-400 e^{-x / 4}\right]_0^8\] Evaluate at the limits: \(-400 e^{-8/4} - (-400 e^{0/4})\).

Step 5: Evaluate the Expression

Calculate the values of the exponential expressions: \(-400 e^{-2} = -400 / e^2\) and \(-400 e^0 = -400\). Thus the evaluation becomes: \(-400 / e^2 + 400\).

Step 6: Calculate the Numerical Answer

Substitute the approximate value of \(e^2\) which is about 7.389 and calculate:\(-400 / 7.389 + 400\). This results in approximately: \(400 - 400 / 7.389 \approx 346.57\).

Key Concepts

Exponential Functions

Exponential functions are a special type of function where the variable appears in the exponent. These functions have the form \( f(x) = a \cdot e^{bx} \) where \( e \) is the base of the natural logarithm, approximately 2.718. In this problem, the function modeling the sales is \( 100 e^{-x/4} \). This indicates that the sales decrease exponentially over time.

Key properties of exponential functions include their rapid growth or decay.

• The base \( e \) makes these functions very applicable in real-world models, such as population growth or radioactive decay.
• Exponential functions are always positive and never touch the horizontal axis, as they approach zero.

Understanding these properties helps in interpreting how the sales will change over the weeks.

Integration Techniques

Integration is the calculus technique used to find the area under a curve, which can represent a total quantity in applied problems. In our exercise, we need to find \(\int_0^8 100 e^{-x / 4} \, dx\)to determine how many condominiums are sold during the first 8 weeks.

For this integral, substitution is a useful method. Setting \( u = -x/4 \) helps simplify the integration process. Here's what happens:

• First, differentiate \( u \) to find \( du \), giving \( du = -1/4 \, dx \).
• Substitute \( dx = -4 \, du \) into the integral to change it in terms of \( u \).

This results in a simpler exponential integral \(-400 \int e^{u} \, du\), which is straightforward to integrate.
Substitution is a common technique used to tackle integrals that aren't immediately apparent.

Calculus Problem Solving

Solving calculus problems involves several strategies, such as interpreting the problem, setting up integrals, and computing solutions step-by-step. Here, understand the role of a definite integral—finding the cumulative total.

In this problem, follow these steps:

• Identify the function that describes the scenario. Here, it is \( 100 e^{-x/4} \), representing sales over time.
• Understand the limits of integration—0 to 8 weeks—determine the interval to calculate.
• Execute substitution for ease of integration, as the direct antiderivative was not readily obvious.
• Evaluate the definite integral at these limits to find the final quantity sold within 8 weeks.

Simplification and approximation wrap up the problem, giving the numerical answer. This problem-solving method underscores the power of calculus in deducing real-world quantities and trends.