Problem 108 - Calculus Volume 1

Question

[T] A condominium in an upscale part of the city was purchased for \(\$ 432,000\) . In 35 years it is worth \(\$ 60,500\) . Find the rate of depreciation.

Step-by-Step Solution

Answer

The rate of depreciation is approximately 8.39% per year.

Step 1: Identify the Depreciation Formula

We will use the formula for depreciation: \[ V = P(1 - r)^t \] where \( V \) is the future value (\\(60,500), \( P \) is the initial value (\\)432,000), \( r \) is the rate of depreciation, and \( t \) is the time in years (35 years).

Step 2: Substitute Known Values

Substitute the given values into the depreciation formula: \[ 60,500 = 432,000(1 - r)^{35} \]. This equation now contains only one unknown variable, \( r \), which we need to solve for.

Step 3: Isolate the Depreciation Rate

Firstly, divide both sides by \( 432,000 \) to start isolating \( r \): \[ \frac{60,500}{432,000} = (1 - r)^{35} \]. This simplifies to approximately \( 0.14 = (1 - r)^{35} \).

Step 4: Take the 35th Root

To solve for \( 1 - r \), take the 35th root of both sides: \[ \sqrt[35]{0.14} = 1 - r \]. Calculating this gives us approximately \( 1 - r = 0.9161 \).

Step 5: Solve for the Rate of Depreciation

Finally, solve for \( r \): \[ r = 1 - 0.9161 \]. Therefore, \( r \approx 0.0839 \), which means the rate of depreciation is \( 8.39\% \).

Key Concepts

Rate of Depreciation

The rate of depreciation is a measure of how quickly the value of an asset decreases over time. In real estate, it indicates how much a property loses its value annually. Understanding this rate is crucial for investors and property owners since it affects an investment's future worth. Depreciation can differ based on various factors including location, property condition, and market trends. In this specific context, the rate tells us the percentage decrease in the property's value each year. To find this rate, we use exponential depreciation formulas, which allow us to calculate the consistent rate at which an asset depreciates over time. By determining the rate of depreciation, individuals can strategize better financial planning and investments.

Depreciation Formula

The depreciation formula is a mathematical expression used to calculate the reduced value of an asset over time. The formula is expressed as \[ V = P(1 - r)^t \] where:

• \( V \) is the asset's value at a later date,
• \( P \) is the initial purchase price,
• \( r \) is the rate of depreciation,
• \( t \) is the time period during which the depreciation occurs.

In this formula, the expression \( (1 - r)^t \) represents the decay factor over the given time period. As time increases, the value of the asset decreases at an exponential rate determined by \( r \). This decay accounts for the asset losing value each year. Understanding this formula helps in predicting the asset's future worth and determining how its value will change over time.

Isolating Variables

To solve for the rate of depreciation in a formula like \[ V = P(1 - r)^t \], we need to isolate the variable of interest, which is \( r \) in this case. Isolating variables involves rearranging the equation to solve for a specific variable. Accurately isolating \( r \) requires careful manipulation of the equation:

• First, divide both sides by the initial value \( P \), to simplify the equation.
• Then, solve the resulting equation for the power \((1 - r)\) by performing operations like taking roots, or utilizing logarithms if necessary.
• Finally, subtract from 1 to isolate \( r \).

Each step requires precision, ensuring variables remain correctly placed on each side of the equation. Isolating variables is an essential algebraic skill that facilitates the solution of complex equations.

Solving Exponential Equations

Solving exponential equations involves dealing with equations where variables appear in the exponent. Such equations require special techniques to unravel. In our depreciation problem, after substituting known values into the formula, we are left with an exponential equation of the form \[ (1 - r)^{35} = 0.14 \]. This indicates that \( 1 - r \) is raised to the power of 35.To solve this, take the 35th root of both sides, effectively reversing the operation of exponentiation. This requires a strong grasp of the properties of exponents and roots:

• Determine the root of the entire expression, which calculates the value of \( 1 - r \).
• Following this, rearrange the equation to isolate \( r \) as explained in the previous section.
• If unknown variables remain within a more intricate equation, logarithms can also be helpful to simplify complex exponential terms.

Mastering these methods equips you with essential tools for solving equations involving exponential growth or decay, like those found when computing depreciation rates.