Problem 25 - Calculus Concepts

Question

Antique Value The monetary value of a certain antique chair increases with its age (but at a diminishing rate). The rate of change in the value of the chair can be modeled as \(v(x)=\frac{2500}{x^{1.5}}\) dollars per year where \(x\) years is the age of the chair, \(x \geq 25 .\) The chair was valued at \(\$ 300\) twenty-five years after it was crafted. a. How much will the value of the antique increase between 25 and 100 years after it was crafted? How much will it be worth 100 years after it was crafted? (Disregard inflation of the dollar.) b. How much will the chair eventually be worth?

Step-by-Step Solution

Answer

The value increases by $500 from 25 to 100 years, worth $800 after 100 years. Eventually, it's worth $1300.

Step 1: Understanding the Rate of Change

The rate of change in the value of the chair is given by the function \( v(x) = \frac{2500}{x^{1.5}} \). This function tells us how much the value increases per year depending on the age \( x \) of the chair, starting from 25 years.

Step 2: Calculating the Increase in Value Between 25 and 100 Years

To find the increase in value between \( x = 25 \) and \( x = 100 \), we need to integrate the rate function \( v(x) \) from 25 to 100. This will give us the total increase in value over this period.\[\int_{25}^{100} \frac{2500}{x^{1.5}} \, dx\]To solve this, we use the formula for the power rule of integration:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]So,\[\int \frac{2500}{x^{1.5}} \, dx = 2500 \int x^{-1.5} \, dx = 2500 \left( \frac{x^{-0.5}}{-0.5} \right) + C = -5000 x^{-0.5} + C\]Evaluate this from 25 to 100:\[\left[ -5000 x^{-0.5} \right]_{25}^{100} = \left( -5000 (100)^{-0.5} + 5000 (25)^{-0.5} \right)\]\[= (-5000 \cdot \frac{1}{10}) + (5000 \cdot \frac{1}{5}) = -500 + 1000 = 500\]So, the value of the chair increases by \( \$500 \) between 25 and 100 years.

Step 3: Determining Value at 100 Years

The chair was valued at \( \\(300 \) at 25 years. It increased by \( \\)500 \) by the time it reaches 100 years. Therefore, the value at 100 years is:\[\300 + 500 = 800 \]The chair will be worth \( \$800 \) 100 years after it was crafted.

Step 4: Calculating the Value as Age Approaches Infinity

To find the eventual value of the chair, we calculate the total increase from 25 years to an infinite age by integrating from 25 to infinity. \[\int_{25}^{\infty} \frac{2500}{x^{1.5}} \, dx\]Apply the limits:\[\lim_{t \to \infty} \left( -5000 x^{-0.5} \right)_{25}^{t} = \lim_{t \to \infty} \left( -5000 (t)^{-0.5} + 5000 (25)^{-0.5} \right)\]\[= 0 + 5000 \cdot \frac{1}{5} = 1000\]The total increase over the infinite time starting from year 25 is \( \\(1000 \). The original value at 25 years was \( \\)300 \). Thus, the chair will eventually be worth \( 300 + 1000 = \$1300 \).

Key Concepts

Rate of Change

The concept of Rate of Change is fundamental in understanding how quickly or slowly something is increasing or decreasing over time.
The mathematical expression of the Rate of Change in this problem is presented as a function, specifically:

• \( v(x) = \frac{2500}{x^{1.5}} \)

This function provides the rate at which the value of the antique chair changes per year. Here, \( x \) represents the age of the chair in years.The function indicates that as the chair ages, the rate at which its value increases decreases. This is evident because the exponent \(-1.5\) results in a decreasing function as \( x \) gets larger.In simple terms, the older the chair gets, the slower its value increases. Understanding this rate helps predict how much the chair's value will grow over specific periods, such as from 25 to 100 years. It's a crucial step in the process of determining eventual values of antiques over time based on increasing age.

Integration in Calculus

Integration in calculus is an essential tool used to calculate accumulated change or total value over a period. In this context, we use integration to find out how much the value of the chair increases over time, between certain ages.To calculate the increase in value from 25 to 100 years, we integrate the function \( v(x) \) from 25 to 100:\[\int_{25}^{100} \frac{2500}{x^{1.5}} \, dx = [-5000x^{-0.5}]_{25}^{100}\]The solution involves applying the power rule of integration. By integrating the rate of change function, we find the total change in value, which turned out to be $500.Integrating the function helps us precisely determine how much the chair’s value grows over defined periods. In this problem, integration provided a better understanding of the incremental increase in the chair's value as it aged from 25 to 100 years, and later as it approaches infinity.

Age and Value Relationship

Understanding the Age and Value Relationship is crucial in problems like the antique value problem.
This relationship defines how the value of an antique changes over time based on its age. In our problem, as the chair gets older, we find the relationship indicating that value growth slows down due to the rate of change function. The chair’s value at age 25 was $300. As it hit 100 years, the total value increased further by $500, making it $800. However, considering an infinite timeline, using calculus, it suggests that the value would eventually stop increasing and consolidate at $1300.

• Initially: $300 when the chair is 25 years old.
• After 100 years: $800 due to the $500 increase.
• Over infinite time: eventually reaches a maximum of $1300.

This relationship between age and value demonstrates the principle that although value increases with age, the increments become lesser over time, eventually reaching a maximum possible valuation.