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Chapter 6: Problem 36

The value of a home, originally worth $$\$ 25,000$$, grows continuously at the rate of $$6 \%$$ per year. Find a formula for its value after \(t\) years.

Short Answer

Expert verified
The value of the home after \( t \) years is \( V(t) = 25000 e^{0.06t} \).

Step by step solution

01

Understand the Continuous Growth Formula

Continuous growth can be modeled with the formula \( V(t) = V_0 e^{rt} \), where \( V_0 \) is the initial value, \( r \) is the growth rate as a decimal, and \( t \) represents time in years. In this problem, the initial value \( V_0 \) is \$25,000, and the growth rate \( r \) is \( 6\% \), or \( 0.06 \) as a decimal.
02

Substitute Known Values into the Formula

Substitute the given values into the continuous growth formula. We have \( V_0 = 25000 \) and \( r = 0.06 \), so the formula becomes \( V(t) = 25000 e^{0.06t} \).
03

Simplify the Formula

This step involves simplification, but the formula \( V(t) = 25000 e^{0.06t} \) is already in its simplest form. This equation shows the relationship between the time \( t \) and the value of the home.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Growth Formula
The continuous growth formula is a powerful tool in understanding how values increase over time. It is expressed as \( V(t) = V_0 e^{rt} \). Here, \( V_0 \) represents the initial value鈥攚hat you start with. The constant \( e \) is the base of natural logarithms and is approximately equal to 2.71828. Meanwhile, \( r \) is the growth rate in decimal form, and \( t \) is time, often in years.

In terms of exponential growth, the continuous growth formula allows for a smooth calculation of growth over any time period. This is particularly useful in real-world scenarios where values do not grow in staggered steps but continuously.
  • The formula is used when changes don't happen in discrete time intervals.
  • The continuous nature of \( e^{rt} \) captures natural growth processes, making it ideal for many applications.
Home Value Appreciation
Home value appreciation is a key concept in real estate, referring to the increase in the market value of a property over time. Continuous growth of home value is a practical scenario in many housing markets across the globe. Appraisal rates, like the 6% annual increase in this exercise, help estimate how much a home might be worth in the future.

Understanding appreciation is essential for both homeowners and investors:
  • Homeowners can estimate future value to plan long-term finances or consider whether to sell.
  • Investors assess property potential and decide on acquiring or retaining real estate for profit.
Using the continuous growth formula gives a mathematical model to predict such changes in value accurately.
Mathematical Modeling
Mathematical modeling is the process of using mathematical structures to represent real-world systems. This allows us to explore scenarios and make informed predictions. When examining how a home's value increases over time, the continuous growth formula is a form of mathematical modeling.

Models simplify reality to help answer questions without needing to account for every minor detail:
  • They provide a simplified analysis of complex systems, often offering insights that are not easily visible.
  • The assumptions in models, like constant growth rates, may not hold exactly but still offer useful approximations.
By transforming a real-life problem into a mathematical formula, we can simulate situations, understand potential outcomes, and make decisions based on quantifiable data.

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Most popular questions from this chapter

A person can memorize at most 40 two-digit numbers. If that person can memorize 15 numbers in the first 20 minutes, find a formula for the number that can be memorized in \(t\) minutes. Use your answer to estimate how long the person will take to memorize 30 numbers.

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=5 y(100-y)\) \(y(0)=10\)

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. \(y^{\prime}=2 y^{2}(0.5-y)\)

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. \(y^{\prime}=0\) \(y(0)=5\)

True or False: If \(f(x)\) is continuous, nonnegative, and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{1}^{x} f(x) d x\) converges.
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